Effects of Curing Temperature and Pressure on the
Chemical, Physical, and Mechanical Properties of
Portland Cement
Xueyu Pang
Submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2011
© 2011
Xueyu Pang
All rights reserved
ABSTRACT
Effects of Curing Temperature and Pressure on the Chemical, Physical,
and Mechanical Properties of Portland Cement
Xueyu Pang
This dissertation mainly focuses on studying the fundamental hydration kinetics and
mechanisms of Portland cement as well as the effects of curing temperature and pressure on its
various properties.
An innovative test apparatus has been developed in this study to cure and test cement paste
specimens under in-situ conditions, such as down-hole in oil wells with high temperature and
high pressure. Two series of tests were performed using cement pastes prepared with four
different classes of oilwell cement (namely Class A, C, G, and H cements). Specimens in groups
of four were cured at temperatures ranging from ambient to 60 °C and pressures ranging from
0.69 to 51.7 MPa for a period of 48 or 72 hours. The density and w/c ratio of the specimens at
the time of casting as well as at the end of the curing period were recorded. Total chemical
shrinkage of the cement paste was measured continuously during the entire hydration period
while tensile strength was obtained at the end of the curing period using both water pressure and
splitting tension test methods. Due to capacity limitations of the test equipment, in-situ tensile
strength was obtained for only one test series with a highest curing pressure of 13.1 MPa.
Specimens from the other test series were depressurized before the tensile strength tests.
Chemical shrinkage test is an important method of measuring cement hydration kinetics in
that the normalized total chemical shrinkage is approximately equal to the degree of cement
hydration. By studying the correlations between the chemical shrinkage and the non-evaporable
water content of cement during hydration, a multi-linear model is first proposed to estimate the
normalization factors for different types of cement under different curing conditions. Based on
the hydration kinetics data obtained from chemical shrinkage test results, a new approach of
modeling the effect of curing temperature and pressure on cement hydration kinetics is proposed.
It is found that when a hydration kinetics curve is represented by an unknown function, the effect
of curing condition on the curve can be modeled by incorporating a simple scale factor in this
function. The relationship between this scale factor and curing condition is described by
chemical kinetics laws.
While the proposed new approach of modeling cement hydration kinetics has the advantage
of being widely applicable to different types of cement, it only explains one influence factor of
cement hydration (i.e. the curing condition). In order to take into account other influence factors
and to further understand the fundamental mechanisms of cement hydration, a more complex
particle-based numerical hydration model is developed by combining the two well-known
cement hydration mechanisms, namely the nucleation and growth controlled mechanism and the
diffusion controlled mechanism. The model is applied to experimental data of both C3S
hydration in dilute suspensions and Class H cement paste hydration. Excellent agreement is
observed between experimental and modeled results. Three rate-controlling parameters with
clear physical meanings can be identified from the proposed model. Fitted model parameters are
found to be in reasonable agreement with experimental observation. The dependencies of these
parameters on particle size, cement composition, w/c ratio, and curing condition are also
investigated.
Finally, the importance of cement hydration kinetics is illustrated by showing their close
correlations with the physical and mechanical properties. The various influence factors,
including the curing temperature and pressure, of physical and mechanical property test results
(particularly density and tensile strength) are evaluated. Potential damage mechanisms of cement
paste specimens during depressurization are studied by analyzing the deformation behavior of
the entire system consisting of the cement paste and pressurizing water.
TABLE OF CONTENTS
CHAPTER 1 : INTRODUCTION .................................................................................................. 1
1.1 Background and Objectives .................................................................................................. 1
1.1.1 Portland cement in the construction and petroleum industries ....................................... 1
1.1.2 Challenges in oilwell cementing..................................................................................... 3
1.1.3 Initiation of the in-situ testing concept ........................................................................... 5
1.1.4 Objectives and scope of the study .................................................................................. 6
1.2 Literature Survey ................................................................................................................... 7
1.2.1 Chemical shrinkage tests ................................................................................................ 8
1.2.2 Hydration kinetics of Portland cement ......................................................................... 10
1.2.3 Tensile strength tests .................................................................................................... 14
1.3 Outline of the Dissertation .................................................................................................. 16
CHAPTER 2 : EXPERIMENTAL METHODS AND PROGRAM ............................................. 19
2.1 Materials and Methods ........................................................................................................ 19
2.2 Test Apparatus and Procedure............................................................................................. 22
2.3 Test Program ....................................................................................................................... 29
2.3.1 Preliminary tests ........................................................................................................... 29
2.3.2 Pressure cell tests .......................................................................................................... 30
2.3.3 Isothermal calorimetry tests.......................................................................................... 33
2.4 Test Data Collection and Analysis ...................................................................................... 33
2.4.1 Test data collection and processing .............................................................................. 33
2.4.2 Influence factors of chemical shrinkage test results ..................................................... 38
2.4.4 Reproducibility of isothermal calorimetry tests ........................................................... 45
CHAPTER 3 : CORRELATION BETWEEN CHEMICAL SHRINKAGE AND THE DEGREE
OF HYDRATION OF CEMENT ................................................................................................. 47
3.1 Introduction ......................................................................................................................... 47
3.2 Classification of Water in Cement Paste ............................................................................. 50
3.3 Mass Fractions of Different Types of Water in Cement Paste............................................ 50
3.4 Specific Volumes of Different Types of Water in Cement Paste ....................................... 55
3.5 Correlation between Chemical Shrinkage and Non-evaporable Water .............................. 61
i
3.6 Model Application............................................................................................................... 63
3.7 Summary ............................................................................................................................. 65
CHAPTER 4 : MODELING THE EFFECT OF CURING TEMPERATURE AND PRESSURE
ON CEMENT HYDRATION KINETICS ................................................................................... 66
4.1 Introduction ......................................................................................................................... 66
4.2 Preliminary Analysis of Test Data ...................................................................................... 68
4.2.1 Chemical shrinkage data ............................................................................................... 68
4.2.2 Hydration kinetics data ................................................................................................. 69
4.3 Model Development ............................................................................................................ 75
4.3.1 Model formulation ........................................................................................................ 75
4.3.2 Significance of the scale factor C(T, P) ........................................................................ 77
4.4 Effect of Curing Temperature on Hydration Kinetics ......................................................... 78
4.5 Effect of Curing Pressure on Hydration Kinetics................................................................ 83
4.6 Verification of the Proposed Model .................................................................................... 86
4.7 Summary ............................................................................................................................. 90
CHAPTER 5 : NUMERICAL MODELING OF CEMENT HYDRATION KINETICS ............ 91
5.1 Introduction ......................................................................................................................... 91
5.2 Theoretical Background ...................................................................................................... 95
5.3. Model Formulation ............................................................................................................. 98
5.3.1 Modeling the nucleation and growth controlled stage.................................................. 99
5.3.2 Modeling diffusion controlled stage ........................................................................... 103
5.3.3 Modeling the total hydration kinetics ......................................................................... 104
5.4 Model Application: C3S Hydration in Dilute Suspensions ............................................... 106
5.4.1 Effects of particle size distribution ............................................................................. 106
5.4.2 Physical meanings of model parameters .................................................................... 113
5.4.3 Effect of the number of initial nuclei.......................................................................... 115
5.4.4 Further discussion ....................................................................................................... 117
5.5 Model Application: Class H Cement Paste Hydration ...................................................... 119
5.5.1 Model modifications and application procedures ....................................................... 119
5.5.2 Effect of curing temperature ....................................................................................... 121
ii
5.5.3 Effect of curing pressure............................................................................................. 126
5.5.4 Effect of w/c ratio ....................................................................................................... 128
5.5.5 Effect of cement composition ..................................................................................... 130
5.6 Implications for Cement Hydration Mechanisms ............................................................. 134
5.7 Summary ........................................................................................................................... 136
CHAPTER 6 : CORELATION BETWEEN CHEMICAL SHRINKAGE AND HEAT OF
HYDRATION OF CEMENT ..................................................................................................... 140
6.1 Introduction ....................................................................................................................... 140
6.2 Preliminary Analysis of Test Data .................................................................................... 142
6.3 Theoretical Analysis .......................................................................................................... 148
6.4 Estimating the Correlation Factors .................................................................................... 151
6.5 Summary ........................................................................................................................... 158
CHAPTER 7 : EFFECT OF CURING TEMPERATURE AND PRESSURE ON THE
PHYSICAL AND MECHANICAL PROPERTIES OF CEMENT............................................ 159
7.1 Introduction ....................................................................................................................... 159
7.2 Results of the Preliminary Tests........................................................................................ 161
7.3 Results of the In-Situ Pressure Cell Tests (Series I) ......................................................... 164
7.4 Results of the Non In-Situ Pressure Cell Tests (Series II) ................................................ 169
7.4.1 Effect of curing conditions on the physical properties of cement .............................. 169
7.4.2 Deformation behavior of cement paste during pressurization and depressurization .. 173
7.4.3 Effect of curing condition on the mechanical properties of cement ........................... 178
7.5 Summary ........................................................................................................................... 200
CHAPTER 8 : CONCLUSIONS AND FUTURE WORK ......................................................... 203
8.1 New Chemical Shrinkage Test for Evaluating Cement Hydration Kinetics ..................... 203
8.2 Effect of Curing Temperature and Pressure on Cement Hydration .................................. 205
8.3 A New Explanation of Cement Hydration Mechanisms ................................................... 207
8.4 Water Pressure Tensile Test of Oilwell Cement ............................................................... 208
8.5 Damage Mechanism of Cement Paste Specimen during Depressurization ...................... 210
8.6 Recommendations for Future Research ............................................................................ 211
iii
LIST OF FIGURES
Figure 1.1: Simplified schematic of an oil well .............................................................................. 4
Figure 1.2: Typical setup of fluid pressure tensile tests.................................................................. 6
Figure 2.1: Particle size distributions of the different types of cement ........................................ 21
Figure 2.2: Sketch of the pressure cell (not to scale) .................................................................... 23
Figure 2.3: Schematic of the test system ...................................................................................... 25
Figure 2.4: Temperature evolutions of the pressure cell and of the specimen ............................. 28
Figure 2.5: Tested specimens cut from cylinders of different heights (178mm front, 305mm back)
....................................................................................................................................................... 30
Figure 2.6: Four repeated tests at curing temperature of 60 °C and curing pressure of 6.9 Mpa (i.e.
Tests 60-II and 60-III) ................................................................................................................... 35
Figure 2.7: Average test results of Test 40-IV before and after smoothing ................................. 35
Figure 2.8: Average test results of Test 60-IV before and after smoothing ................................. 36
Figure 2.9: Test plot of a specimen cured at 0.69 Mpa and 24 °C................................................ 37
Figure 2.10: System deformation tests performed at different pressures ..................................... 39
Figure 2.11: Apparent bulk modulus of water at different pressures ........................................... 40
Figure 2.12: Four repeated tests of 24-II/24-III (one with impermeable rubber sleeves) (Class HI cement, w/c=0.4) ........................................................................................................................ 42
Figure 2.13: Four repeated tests of 24-I (Class H-II cement, w/c=0.4) ........................................ 42
Figure 2.14: Effect of specimen thickness on test results ............................................................. 44
Figure 2.15: Effect of specimen thickness on test results (Calibrated to a uniform temperature of
25.6 °C) ......................................................................................................................................... 45
Figure 2.16: Repeated isothermal calorimetry tests at 25 °C ....................................................... 46
Figure 3.1: Non-evaporable water content vs. C3A content of cement ........................................ 53
Figure 3.2: Variation of specific volume of hydrated cement paste with evaporable water content
....................................................................................................................................................... 59
Figure 3.3: Total volume change during cement hydration .......................................................... 62
Figure 4.1: Representative hydration kinetic curves of Class H-II cement (w/c = 0.38) ............. 66
Figure 4.2: Effect of curing temperature and pressure on total chemical shrinkage (Class H-II
cement, w/c = 0.4)......................................................................................................................... 68
Figure 4.3: Effect of curing pressure on total chemical shrinkage (Class H-II cement, w/c = 0.38,
ambient temperatures) ................................................................................................................... 69
Figure 4.4: Effect of curing temperature on cement hydration kinetics (Class H-II cement, w/c =
0.4, curing pressure = 13.1 MPa) .................................................................................................. 70
Figure 4.5: Effect of curing pressure on cement hydration kinetics (Class H-II cement, w/c =
0.38, ambient temperatures) .......................................................................................................... 71
Figure 4.6: Effect of curing temperature on hydration rate as a function of degree of hydration
(Class H-II cement, w/c = 0.4, curing pressure = 13.1 MPa) ....................................................... 72
Figure 4.7: Effect of curing pressure on hydration rate as a function of degree of hydration
(Class H-II cement, w/c = 0.38, ambient temperatures) ............................................................... 72
iv
Figure 4.8: Effect of curing pressure on hydration rate as a function of degree of hydration
(Class C cement, w/c = 0.56, ambient temperatures) ................................................................... 74
Figure 4.9: Normalized hydration rate as a function of degree of hydration for different cement
(w/c ratios of Class A, C, G, H cement are 0.46, 0.56, 0.44, and 0.38, respectively) .................. 74
Figure 4.10: Variation of activation energy with degree of hydration ......................................... 80
Figure 4.11: Linear regression analyses showing the temperature dependence of the scale factor
C(T) for different cements ............................................................................................................ 83
Figure 4.12: Linear regression analyses showing the pressure dependence of the scale factor C(P)
for different cement....................................................................................................................... 84
Figure 4.13: Variation of activation volume with degree of hydration ........................................ 86
Figure 4.14: Measured and predicted hydration kinetics of different types of cement cured at
different temperatures (Ambient condition as the reference) ....................................................... 88
Figure 4.15: Measured and predicted hydration kinetics of different types of cement cured at 51.7
MPa (0.69 MPa curing pressure as the reference) ........................................................................ 89
Figure 4.16: Measured and predicted hydration kinetics of different types of cement cured at 51.7
MPa (0.69 MPa curing pressure as the reference) ........................................................................ 89
Figure 5.1: Schematic of assumed C3S hydration mechanism during NG stage ....................... 100
Figure 5.2: The flowchart of computer simulation of the hydration of a single C3S particle..... 106
Figure 5.3: Experimental (Garrault 2006) and modeled results of degree of hydration of samples
with different particle sizes ......................................................................................................... 108
Figure 5.4: Cumulative particle size distribution curves for the five samples (Garrault 2006).. 109
Figure 5.5: Effect of initial particle size on hydration kinetics................................................... 110
Figure 5.6: Modeled rate of hydration of samples with different particle sizes ......................... 110
Figure 5.7: Comparison between modeled curves obtained with single particle size vs. multiple
particle sizes ................................................................................................................................ 112
Figure 5.8: Effect of different parameters on degree of hydration curve ................................... 115
Figure 5.9: Experimental (Garrault 2001) and fitted hydration kinetics of C3S in saturated lime
solution with different quantities of C-S-H nuclei ...................................................................... 116
Figure 5.10: Example of fitting the proposed model to experimental data of Test H-II-5-1 ...... 121
Figure 5.11: Experimental and modeled hydration kinetics of Class H-II cement (w/c = 0.4,
curing pressure = 13.1 MPa) ....................................................................................................... 122
Figure 5.12: Experimental and modeled hydration kinetics of Class H-II cement cured at
different temperatures and pressures (w/c = 0.4) ........................................................................ 122
Figure 5.13: The temperature dependence of the scale factors (Class H-II cement, w/c = 0.4) . 125
Figure 5.14: Experimental and modeled hydration kinetics of Class H-II cement cured at
different pressures and at ambient temperatures (w/c = 0.38) .................................................... 126
Figure 5.15: The pressure dependence of the scale factors (Class H-II cement, w/c = 0.38) ..... 128
Figure 5.16: Experimental and modeled hydration kinetics of Class H-II cement with different
w/c ratios and cured at different pressures .................................................................................. 130
v
Figure 5.17: Experimental and modeled hydration kinetics of Class H-P cement cured at
temperatures and pressures (w/c = 0.38) .................................................................................... 132
Figure 5.18: The temperature and pressure dependence of the scale factors (Class H-P cement,
w/c = 0.38) .................................................................................................................................. 133
Figure 5.19: Modeled hydration kinetics of individual cement particles and the weighted average
result of a sample with multiple particle sizes ............................................................................ 136
Figure 6.1: Effect of curing temperature on hydration rate as a function of degree of hydration
(Class H-I cement, w/c = 0.38) ................................................................................................... 144
Figure 6.2: Normalized differential equation curves of different types of cement ..................... 144
Figure 6.3: Measured and predicted hydration kinetics at different curing temperatures by
coordinate transformations (Class C cement, w/c = 0.56) .......................................................... 146
Figure 6.4: Measured and predicted hydration kinetics of different cements at different curing
temperatures by coordinate transformations ............................................................................... 147
Figure 6.5: Heat evolution curves vs. transformed chemical shrinkage curves (before offset).. 152
Figure 6.6: Heat evolution curves vs. transformed chemical shrinkage curves (after offset) .... 153
Figure 6.7: Heat evolution curves vs. transformed chemical shrinkage curves for different types
of cement ..................................................................................................................................... 154
Figure 6.8: Heat evolution curves vs. transformed chemical shrinkage curves for different types
of cement at different curing temperatures ................................................................................. 155
Figure 6.9: Dependence of correlation factor (CS0/H0) on curing temperature .......................... 157
Figure 7.1: Effect of bearing strip width on splitting tensile strength ........................................ 162
Figure 7.2: Splitting tensile strength variation along vertical direction of cylindrical samples . 163
Figure 7.3: Typical locations of fracture planes under hydraulic pressure ................................. 164
Figure 7.4: Comparison between test results of Class H-I and H-II cements ............................. 166
Figure 7.5: Effect of curing condition on splitting tensile test results (Class H-II cement, w/c =
0.4, age = 48 h) ........................................................................................................................... 167
Figure 7.6: Effect of curing condition on in-situ water pressure test results (Class H-II cement,
w/c = 0.4, age = 48 h) ................................................................................................................. 167
Figure 7.7: Average splitting tensile strength vs. average water-pressure tensile strength (Class
H-II cement, w/c = 0.4, age = 48 h) ............................................................................................ 168
Figure 7.8: Dependence of specimen density on curing condition (Class A cement, w/c = 0.46,
age = 72 h) .................................................................................................................................. 172
Figure 7.9: Dependence of specimen density on effective w/c ratio of different cement .......... 172
Figure 7.10: Pressure and volume variations with time of a system mainly consists of cement
paste and pressurizing water ....................................................................................................... 174
Figure 7.11: Variation of total system volume with pressure at different ages .......................... 175
Figure 7.12: Variation of system bulk modulus with pressure ................................................... 176
Figure 7.13: Variation of system deformation gradient with pressure ....................................... 177
Figure 7.14: Effect of curing temperature on the splitting tensile strength of cement (age = 72 h,
curing pressure = 0.69 MPa) ....................................................................................................... 179
vi
Figure 7.15: Effect of curing temperature on the water pressure tensile strength of cement (age =
72 h, curing pressure = 0.69 MPa) .............................................................................................. 180
Figure 7.16: Effect of curing pressure on the hydration kinetics of different cements (ambient
curing temperatures) ................................................................................................................... 181
Figure 7.17: Representative fractured specimens after water pressure tests (Class H-II cement)
..................................................................................................................................................... 182
Figure 7.18: Fractured specimens after splitting tensile tests (Class H-II cement) .................... 183
Figure 7.19: All specimens of Test H-II-4 after water pressure tests ......................................... 183
Figure 7.20: Variation of system deformation gradient with pressure (Class H-I and H-II
cements, w/c = 0.38) ................................................................................................................... 184
Figure 7.21: Variation of system deformation gradient with pressure (Class H-II cement)....... 185
Figure 7.22: Effect of curing pressure on the tensile strength of Class H-I and H-II cement (w/c
= 0.38, age = 72 h) ...................................................................................................................... 186
Figure 7.23: Tensile strength of Class H-II cement with different w/c ratios (age = 72 h) ........ 187
Figure 7.24: Fracture planes of water pressure tests (left) and splitting tensile tests (right) (Test
H-P-3) ......................................................................................................................................... 188
Figure 7.25: System deformation gradient and bulk modulus variations with pressure (Class H-P
cement, w/c = 0.38)..................................................................................................................... 189
Figure 7.26: Effect of curing condition on the tensile strength of Class H-P cement (age = 72 h,
w/c = 0.38) .................................................................................................................................. 190
Figure 7.27: Variation of system deformation gradient with pressure (Class C cement, w/c = 0.56)
..................................................................................................................................................... 191
Figure 7.28: Effect of curing condition on the tensile strength of Class C cement (age = 72 h,
w/c = 0.56) .................................................................................................................................. 192
Figure 7.29: Fracture planes of water pressure tests (Class A cement) ...................................... 193
Figure 7.30: Fracture planes of splitting tensile tests (Class A cement) ..................................... 194
Figure 7.31: Variation of system deformation gradient with pressure (Class A cement, w/c = 0.46)
..................................................................................................................................................... 195
Figure 7.32: Effect of curing condition on the tensile strength of Class A cement (age = 72 h, w/c
= 0.46) ......................................................................................................................................... 196
Figure 7.33: Representative fractured specimens after water pressure tests (Class G cement) .. 197
Figure 7.34: Fractured specimens after splitting tensile tests (Class G cement) ........................ 198
Figure 7.35: Variation of system deformation gradient with pressure (Class G cement, w/c = 0.44)
..................................................................................................................................................... 199
Figure 7.36: Effect of curing condition on the tensile strength of Class G cement (age = 72 h,
w/c = 0.44) .................................................................................................................................. 200
vii
LIST OF TABLES
Table 1.1: Shorthand notations in cement chemistry ...................................................................... 2
Table 1.2: Experimental methods to measure non-evaporable water content .............................. 13
Table 2.1: Oxide analysis results of the different types of cement ............................................... 20
Table 2.2: Estimated main compound compositions of the different types of cement ................. 20
Table 2.3: List of number of specimens tested at different conditions ......................................... 30
Table 2.4: Pressure cell tests (Series I, w/c = 0.4, Test age = 48 hours) ...................................... 31
Table 2.5: Pressure cell tests (Series II, Test age = 72 hours) ...................................................... 32
Table 2.6: Isothermal calorimetry tests (Atmospheric pressure, Test age = 168 hours) .............. 33
Table 3.1: Total chemical shrinkage at complete hydration of different clinker phases .............. 49
Table 3.2: Coefficients for total non-evaporable water content (P-dried samples) ...................... 53
Table 3.3: Experimental and predicted value of wn0 for different cements .................................. 55
Table 3.4: Estimated values of CS0 for different cement under different curing conditions ........ 64
Table 4.1: Scale factors for different curing conditions and estimated activation energy ............ 82
Table 4.2: Scale factors for different curing conditions and estimated activation volume........... 85
Table 4.3: Summary of the universal model to predict the hydration kinetics curves.................. 87
Table 5.1: Model Parameters for different particle size brackets ............................................... 108
Table 5.2: Model Parameters corresponding with different characteristic particle sizes ........... 117
Table 5.3: Dependence of model parameters on curing condition (Class H-II cement, w/c =0.4)
..................................................................................................................................................... 123
Table 5.4: Scale factors on nuclei growth rate derived from fitted parameters .......................... 125
Table 5.5: Dependence of model parameters and scale factors on curing pressure (Class H-II
cement, w/c =0.38)...................................................................................................................... 127
Table 5.6: Dependence of model parameters and scale factors on w/c ratio (Class H-II cement)
..................................................................................................................................................... 130
Table 5.7: Model parameters and scale factors for Class H-P cement (w/c = 0.38)................... 132
Table 6.1: Coefficients for total heat of hydration H0, in J/g (Taylor 1997a)............................. 142
Table 6.2: Activation energies obtained from different methods ............................................... 148
Table 6.3: Best-fit parameters and estimated temperatures of chemical shrinkage tests............ 156
Table 7.1: Physical properties of specimens from different tests ............................................... 170
viii
ACKNOWLEDGEMENTS
First and foremost, I would like to express my sincere appreciation and gratitude to my
mentor, Professor Christian Meyer, for four years of guidance, support, and encouragement. I
feel very fortunate and truly honored to have Prof. Meyer, a patient and caring educator, a
passionate and devoted scholar, as my advisor during my graduate study at Columbia University.
I am deeply indebted to Dr. Gary Funkhouser, Mr. Robert Darbe, and Mr. David Meadows
of Halliburton Energy services (HES) for their countless help in designing and building the test
apparatus, providing experimental materials and equipment, as well as characterizing the cement
properties. The financial support from HES is cordially appreciated. The continuous support
from Dr. Lewis Norman and Dr. Ron Morgan deserves special mention here. The hospitality,
generosity, and care of many other HES personnel during my several visits to Duncan
Technology Center are also greatly appreciated.
My sincere gratitude is extended to Dr. Dale Bentz of the National Institute of Standards
and Technology for his enthusiastic help in obtaining the isothermal calorimetry test data and
insightful discussions about the test results.
I am very grateful to Professors George Scherer, Raimondo Betti, Huiming Yin, and SiuWai Chan for their interest in my research and their willingness to serve on the defense
committee and review this dissertation. I would like to express my sincere appreciation and
gratitude to Prof. George Scherer for his invitation to Princeton University and his insightful
questions and suggestions, which helped to further improve the quality of this work.
I would also like to thank the faculty of the School of Engineering and Applied Science at
Columbia University for teaching and mentoring me. Prof. Rene Testa deserves my special
thanks for his help in my professional development. My special gratitude also goes to the
ix
Carleton Lab managers, Mr. Adrian Brugger and Dr. Liming Li, for their enormous support of
my experimental work. The countless help from the departmental staff Ms. Elaine Macdonald
and Ms. Christine Persaud in material and equipment purchases and other matters are gratefully
acknowledged.
I must also acknowledge the assistance and friendship of many colleagues at Columbia
University including Kyu Hyuk Kyung, Zheyuan Chen, Jie Yin, Chunmei Qiu, Manman Kang,
Meng Yan, Yue Zhang, Huijie Lu, Shuo Zhang, Donna Chen, Congcui Tang, Xi Yu, Xin Li, Ben
Leshchinsky, Dan Hochstein, Ya-Chuan Huang, Baoxing Xu, Xia Liu, Jie Xu, Jianhong Jiang,
Jianbin Zhao, Yuye Tang, and Ling Liu. Their company has made my study at Columbia much
more enjoyable. Special mention is made to undergraduate student assistant Ms. Donna Chen
who helped me conducting a significant portion of the experimental work.
Finally, I would like to express my most sincere appreciation to my family for their love,
care, and sacrifice. I can hardly imagine myself accomplishing anything without your selfless
devotion and support. To my parents, words are incapable of adequately describe my
gratefulness for all you have done for me. Thank you for always being there for me. To my sister,
I can never thank you enough, for your love, encouragement and guidance throughout my life.
x
Dedicated to my late grandparents for their love.
xi
1
CHAPTER 1 : INTRODUCTION
1.1 Background and Objectives
1.1.1 Portland cement in the construction and petroleum industries
The invention of modern structural concrete has led to its widespread application in
virtually every corner of the world, making it the most widely used man-made material today
(Lomborg 2001). Portland cement, invented in the early 1840s, is the most common type of
binder used in producing concrete. It is estimated that about 3.3 billion tons of Portland cement
were produced worldwide in the year 2010 (U.S. Geological Survey 2011). Cement-based
materials gain their strength and other properties through a process known as hydration, which
involves a number of different chemical reactions occurring simultaneously. Advancements of
cement and concrete technology have helped to dramatically improve the various physical and
mechanical properties of the material. However, due to their complexity, many detailed features
of cement hydration process are still not clearly understood today. A more complete
understanding of cement hydration process holds the prospect of further improving the
performance of cement-based materials.
Portland cement is made from various raw materials containing primarily lime, silica,
alumina and iron oxide. These materials interact with each other during the production process
and form a series of complex compounds, which mainly include tricalcium silicate, dicalcium
silicate, tricalcium aluminate, and tetracalcium aluminoferrite. The chemical formulae of these
compounds are traditionally written in shorthanded oxide notation frequently used by cement
chemists. A list of the most commonly used shorthand notations is shown in Table 1.1 (Neville
1996, Taylor 1997a, Mindess 2002). Direct analysis of compound composition in Portland
2
cement includes microscope examination, X-ray powder diffraction, and scanning electron
microscopy. However, due to high equipment cost and complicated calibration and calculation
processes, the compound composition is usually estimated by calculation using the ideal
compound stoichiometries and an oxide analysis which is usually determined by X-ray
fluorescence spectroscopy (Neville 1996). The calculation of the phases from the composition is
known as the Bogue calculation.
Table 1.1: Shorthand notations in cement chemistry
Oxide
Compound
Formula Abbreviation
Formula
Abbreviation
Common Name
CaO
C
3CaO·SiO2
C3 S
Tricalcium silicate
SiO2
S
2CaO·SiO2
C2 S
Dicalcium silicate
Al2O3
A
3CaO·Al2O3
C3 A
Tricalcium aluminate
Fe2O3
F
4CaO·Al2O3·Fe2O3
C4AF
Tetracalcium aluminoferrite
CaSO4·2H2O
Gypsum
SO3
S
CSH
H2O
H
Portland cement is usually classified into different types according to their chemical
compositions and particle sizes (or specific surface areas). The most well-known and most
widely-used classification in the United States is specified in ASTM C150/C150M (2011). The
five major types of Portland cement are designated Types I through V. The main focus of this
study is a special category of cement used in the petroleum industry, known as oilwell cements.
The American Petroleum Institute writes specifications for cements used only in oil and gas
wells. API specification 10A (2010) specifies eight classes of oilwell cements, designated
Classes A through H. Oilwell cements are based upon Portland cement, but manufactured to a
higher level of consistency from one production batch to another. API Classes A, B, and C
cements, intended for use from surface to 1800m depth in wells, are similar to ASTM Types I, II,
3
and III cements respectively. The differences between oilwell cements and ordinary Portland
cements are not substantial. Therefore, fundamental theories developed for regular Portland
cements are also applicable to oilwell cements, and vice versa.
1.1.2 Challenges in oilwell cementing
Cementing is one of the most important procedures during oil or gas well construction, the
purpose of which is to create a cement sheath in the annulus between the steel casing and the
wellbore. Figure 1.1 is a simplified schematic of an oil well. The cement sheath primarily acts as
a seal to isolate different zones of the formations and prevent the migration of hydrocarbons or
water from one layer to another, i.e. the so-called zonal isolation. In addition, the cement sheath
also serves to protect the casing from corrosion as well as from shock loads in drilling deeper
(Smith 1990). To function properly, oilwell cements must meet certain physical and mechanical
property requirements. As both temperature and pressure increase with the depth of the wellbore,
oilwell cements are subject to wide ranges of temperature and pressure. It is quite a challenge to
study the effects of these extreme curing conditions on the properties of oilwell cements. The
highest temperature encountered in deep oil wells has increased from about 200 °C to 400 °C in
the past century (Ostroot 1964, Eilers 1979, Roy 1979, Gallus 1979, Roy 1980, Eilers 1983)
while the highest pressure is now up to 150 MPa (Labibzadeh 2010, Scherer 2010). It is known
that neat Portland cement pastes experience significant strength retrogression and permeability
increase at temperatures around 110-120 °C [230-247 °F] due to the conversion of its main
hydration product, C-S-H gel, to other crystalline forms (Eilers 1974, Eilers 1983, Zhang 2008).
As a result, different additives have to be used to stabilize the strength and maintain low
permeability of the material under elevated temperatures, further complicating the already
4
complex cement hydration chemistry. Although non-destructive tests using ultrasonic wave
method are widely used today to evaluate the in-situ properties of oilwell cement (Rao 1982, API
1997, Pedam 2007, Reddy 2007a, Labibzadeh 2010), the reliability of these test results depends
on empirical correlations between wave velocities and mechanical properties determined from
destructive tests. Traditional destructive test methods do not allow the properties of cement to be
determined in-situ, i.e. under conditions of high temperature and high pressure. Therefore,
specimens cured under simulated down-hole conditions (or field conditions) have to be tested
after they had been returned to ambient temperature and pressure (Swayze 1954, Oyefesobi 1976,
Oyefesobi 1977, Kukacka 1981, Sasaki 1986, Degouy 1990). Since the disruptions introduced by
the environment change (especially pressure) are difficult to evaluate, it is not clear to what
extent the results from traditional destructive tests can be relied on.
Figure 1.1: Simplified schematic of an oil well
5
1.1.3 Initiation of the in-situ testing concept
The method of fluid pressure testing of concrete originated almost a century ago, when
Bridgman (1912) reported that a cylindrical specimen appeared to have a tensile fracture
transverse to its axis if appropriate fluid pressure was applied to its bare curved surface. The test
method has been further explored and validated by Clayton (1978, 1980) and Mindess et al.
(2005) in recent years. Lile et al. (1997) also used a somewhat similar approach to obtain the
tensile strength of a hydrating cement paste at the age of several hours, by pumping water into
the cement paste cast in a cylindrical cell through a pipe embedded in the specimen. In a typical
Bridgman-type test, a cylindrical concrete specimen is placed in an open-ended steel jacket
sealed with O-rings. A sketch of the testing device is shown in Figure 1.2. The specimen is
fractured by gradually increasing the fluid pressure on the curved surface of the specimen. This
particular test method opened an opportunity for in-situ measurement of the tensile strength of
cement paste cured under high pressures. Based on the same concept, a cylindrical specimen
cured under hydrostatic pressure may be tested by increasing the pressure difference between the
annular zone and end zones. This innovative in-situ test method was first proposed by Meyer in
2005 and further developed by Gary et al. (2009), which resulted in a US patent.
6
Figure 1.2: Typical setup of fluid pressure tensile tests
1.1.4 Objectives and scope of the study
This dissertation centers around an innovative apparatus developed to obtain the water
pressure tensile strength of oilwell cements under in-situ conditions, i.e. without returning the
specimens to ambient temperature and pressure prior to testing. Due to limitations of the
equipment capacities, this study will focus on validating the new tensile strength test method,
rather than simulating the actual downhole conditions. The newly developed test apparatus also
allows cement chemical shrinkage, which is closely related to hydration kinetics, to be measured
continuously under different temperatures and pressures. These chemical shrinkage data can be
used to further investigate the hydration mechanisms of Portland cement. The main objectives of
this study include:
•
Evaluate the appropriateness of using water pressure tests to measure the in-situ tensile
strength of oilwell cement and study the influencing factors of test results.
7
•
Compare the tensile strength test results obtained in-situ with those obtained not in-situ and
identify the damage mechanisms of oilwell cement specimens during the process of pressure
release and propose methods to avoid or minimize such damages.
•
Investigate the correlations between chemical shrinkage and the degree of hydration of
cement and use chemical shrinkage test results to evaluate and model the effect of curing
temperature and pressure on cement hydration kinetics.
•
Compare cement hydration kinetics measured by chemical shrinkage and by isothermal
calorimetry and establish their correlation at different curing temperatures.
•
Develop new, simple models to simulate cement hydration with parameters that have clear
physical meanings and, by fitting the model to experimental data, propose more detailed
explanations of cement hydration mechanisms.
The temperature range studied here is only from ambient (24±2.8 °C) to 60 °C. The model
developed based on this study may allow extrapolations up to a range from 0 to about 100 °C,
since it is believed that in this range temperature has an influence only on the rate, not the nature,
of the hydration process (Scherer 2010). The pressure range investigated in this study varied
from 0.69 to 51.7 MPa (100 to 7500 psi).
1.2 Literature Survey
Before introducing the new test methods developed in this study, it is important to review
the current practices of characterizing the various properties of cement-based materials such that
the different test methods can be compared with each other. Specifically, we will focus on three
main properties of oilwell cement, namely chemical shrinkage, hydration kinetics, and tensile
strength. The first two properties are closely related to each other in that chemical shrinkage is an
8
important process that permits evaluating hydration kinetics. Modeling hydration kinetics is
important to both oilwell cement and ordinary Portland cement because it can be used to predict
many time-dependent properties of cement-based materials. The cause of autogenous shrinkage,
which is critical to zonal isolation, can also be ultimately related to chemical shrinkage and
degree of hydration (Lin 2006). Tensile strength is one of the most important mechanical
properties of cement-based materials because it is closely related the material’s susceptibility to
cracking, which is the direct cause of material failure under stress.
1.2.1 Chemical shrinkage tests
Under atmospheric pressure, chemical shrinkage is measured by monitoring either the
volume or the weight of the water uptake of a thin layer of cement slurry placed in a flask or a
glass vial (ASTM C1608, 2007). Despite the apparently simple principle, there are a lot of
experimental difficulties that may result in spurious results. For example, the test involves
adding a significant amount of water on top of a thin specimen (< 10 mm) to keep it saturated. A
recent study showed that the quantity and the composition of the surface water have a significant
impact on test results (Sant 2006). Increasing the amount of surface water was found to increase
the chemical shrinkage rate before the end of the induction period and reduce the peak chemical
shrinkage rate during later periods. The initial increase is probably due to the accelerated
dissolution as a result of more Ca2+ leaching into the solution. Massive precipitations of large
portlandite crystals were confirmed on the surface of the alite (the main composition of Portland
cement) specimens used for chemical shrinkage measurement (Costoya 2008). Additionally,
when the same measuring device was used, increasing specimen thickness was consistently
found to cause a reduction in chemical shrinkage at later ages (>15 h) (Geiker 1983, Sant 2006,
9
Costoya 2008), which may be explained by two hypotheses: one is that the reduction in the
permeability of the specimen prevented surface water from filling all the pores in the thicker
specimens; the other is that a larger fraction of the thinner specimen was diluted by the surface
water, resulting in a faster hydration rate at later ages. However, for the same amount of alite
paste, Costoya found that using a small diameter device with less surface water (cylindrical flask)
systematically gave a higher chemical shrinkage than using a large diameter device with more
surface water (Erlenmeyer flask) even though the former generated a much thicker specimen.
The author also found that chemical shrinkage measured with the former device was the same as
that measured with a set ground paste specimen for a period exceeding 250 hours. Therefore, the
surface water probably has a much stronger effect on test results than the thickness of the
specimen and the latter is probably not a limiting factor.
Chemical shrinkage of Portland cement is believed to be directly related to its degree of
hydration (Parrott 1990, Bentz 1995). When chemical shrinkage tests are used to study the
hydration kinetics of cement, it is necessary to automate the data collection process and several
different techniques have been used for such purpose (Geiker 1983, Mounanga 2006,
Peethamparan 2010). However, test results of these studies typically do not allow reliable
generation of the derivative curves (i.e. rate of chemical shrinkage as a function of time), which
correspond with hydration rate curves, due to data oscillations. Mounanga et al. (2006) adopted
polynomial fits to derive the derivative curves while Zhang (2010) showed that a derivative
curve can be obtained by applying a cement hydration model (Thomas 2009) to test data. Sant et
al. (2008) were among the first to derive relatively reliable derivative curves (without excessive
oscillation) of chemical shrinkage directly from experimental data.
10
Justnes et al. (1995) studied the chemical shrinkage of neat Class G oilwell cement slurry
using the traditional flask method. The amount of shrinkage at 20 °C and atmospheric pressure
was found to be about 2.2 ml/100 g cement after 48 hours. Early studies of oilwell cement
shrinkage under pressure either used sealed specimens (Chenevert 1991) or involved equipment
with pistons pushing directly against cement pastes (Parcevaux 1984, Jennings 2005). These
results did not represent the total chemical shrinkage as the internal voids in the specimens
formed during hydration were not accounted for. It is well known that the total chemical
shrinkage of cement increases with increasing curing temperature due to accelerated hydration
rate (Mounanga 2006). Reddy et al. (2007b) were among the first to measure the true chemical
shrinkage of cement under hydrostatic pressure using a water injection pump. However, only
some qualitative analysis of the results can be performed due to the limited number of tests. Most
studies have reported that application of hydrostatic pressure increases the degree of hydration of
cement (Rahman 1982, Bresson 2002, Zhou 2003, Meducin 2007, Scherer 2010). Therefore,
chemical shrinkage of cement can also be expected to increase with increasing pressure as it is
approximately proportional to the degree of hydration.
1.2.2 Hydration kinetics of Portland cement
Although many detailed features of the cement hydration process are still not clearly
understood today, the general hydration kinetics can be approximately represented by the overall
degree of hydration as a function of time. This overall degree of cement hydration, defined as the
total weight fraction of cement reacted, is directly related to many different physical and
mechanical properties of cement-based materials, such as viscosity (Scherer 2010), setting time
(Pinto 1999, García 2008, Zhang 2010), autogenous shrinkage (Lin 2006), compressive strength
11
(Kjellsen 1991, ASTM C1074), tensile strength (Krauss 2006), and modulus of elasticity (Krauss
2006, Lin 2006). It is arguably the most important parameter that can be used to model the timedependent characteristics of cement-based materials (Pane 2002). Since Portland cement mainly
consists of four clinker phases, its overall degree of hydration can be written as (Parrott 1990):
α ( t ) = pC Sα C S ( t ) + pC Sα C S ( t ) + pC Aα C A ( t ) + pC AFα C AF ( t )
3
3
2
2
3
3
4
4
(1.1)
where pi is the original weight fraction of Phase i in the anhydrous cement and αi(t) is the degree
of hydration of Phase i at time t. Direct determination of αi(t) can be made by using quantitative
X-ray diffraction analysis (Parrott 1990, Escalante-Garcia 1998).
Some properties of a hydrating cement paste, such as the non-evaporable water content, the
cumulative heat evolution and the total chemical shrinkage have been shown to have
approximately linear relationships with each other and the overall degree of hydration (Parrott
1990, Bentz 1995, Escalante-Garcia 2003, Zhang 2010). Measuring these properties serves as
alternative ways of determining α, i.e. the so-called indirect methods. As a matter of fact, α is
more commonly determined by these indirect methods due to their simplicity. The following
equation may be used to convert experimental results (obtained from indirect methods) to the
degree of hydration of cement, α(t):
α (t ) =
H (t ) CS (t ) wn (t )
=
=
H0
CS 0
wn0
(1.2)
where H(t) and H0 are the amounts of cumulative heat evolution at time t and at complete
hydration, respectively (typically in J/g cement); CS(t) and CS0 are the amounts of chemical
shrinkage at time t and at complete hydration, respectively (typically in mL/g cement); while
wn(t) and wn0 are the non-evaporable water contents at time t and at complete hydration,
respectively (typically in g/g cement). Apparently, the parameters at the complete hydration
12
condition (namely H0, CS0 and wn0) are essential for experimental data conversion. Different
methods of estimating these parameters as well as the parameters’ correlations with each other
are discussed in Chapters 3 and 6. The hydration reactions of different phases in Portland cement
are known to have different contributions toward the overall parameters (i.e. H(t), CS(t), and
wn(t)). Since these reactions progress at different rates (that also vary with time), the indirect
methods only give a gross approximation to the total hydration kinetics of cement.
The heat evolution during cement hydration is typically measured by monitoring the
thermal output from a specimen kept at near isothermal conditions (isothermal calorimetry).
Standard test procedures are described in ASTM C1679. The limitations of the traditional
chemical shrinkage tests (ASTM C1608) have been discussed in detail in the previous section.
An innovative method that minimizes these limitations and allows chemical shrinkage to be
measured continuously at different temperatures and pressures is used in this study. The methods
to determine the non-evaporable water content of a hydrated cement sample has not been
standardized. Therefore, slightly different results may be obtained with different test methods,
which should be kept in mind when test results from different sources are compared. The
commonly used methods include Loss on Ignition (LOI) test and thermogravimetric analysis
(TGA). Popular drying methods (pretreatment) to remove the evaporable water in the sample
include P-drying (drying over a mixture of di- and tetra-hydrates of magnesium perchlorate at
23 °C), D-drying (drying in an environment of dry ice at -78.5 °C) and oven drying at or slightly
above 100 °C. As it may take a very long time to complete the drying process (i.e. until the
sample reaches constant weight), methanol is sometimes used to stop the hydration to determine
non-evaporable water content at early ages (Parrott 1990). The water-vapor pressure during Ddrying (0.67 Pa) is about 16 times lower than during P-drying (10.7 Pa), which may result in
13
about 8% less non-evaporable water for the former (Brouwers 2004). Powers and Brownyard
(1946) had found that the water-vapor pressure maintained by anhydrous magnesium perchlorate
was much lower than that of its hydrates, and the former resulted in 5-6% less non-evaporable
water; they had also shown that oven drying at 105 °C resulted in about 11% less non-evaporable
water compared with P-drying. Parrott et al. (1990) found that the LOI results are 5% and 7%
higher than the TGA results based on tests conducted by the British Cement Association (BCA)
and those conducted at the Technical University of Denmark (DK), respectively. A summary of
previously used methods to determine the non-evaporable content is provided in Table 1.2.
Table 1.2: Experimental methods to measure non-evaporable water content
Method
LOI
TGA
Source
Pretreatment
Ignition temperature
/Temperature rangea
Powers 1946
P-dryingb
1000 °C
Copeland 1960
P-drying & D-drying
Not specified
Lerch 1948
P-drying
Not specified
Mills 1966
Oven drying at 110°C
1000 °C
Parrott 1990 (BCA)
Methanol, dry silica gel
900 °C
Parrott 1990 (DK)
Methanol, evacuation,
oven drying at 105°C
900 °C
Bentz 1995
Oven drying at 105°C
950 °C
Parrott 1990 (BCA)
Methanol, dry Silica Gel 100-750 °C (2)
Parrott 1990 (DK)
Methanol, evacuation,
oven drying at 105°C
100-700 °C (10)
Mounanga 2004
None
145-1050 °C (10)c
Escalante-Garcia 2003 Vacuum desiccator
150-900 °C (20)
: Ignition temperature for LOI tests or temperature range for TGA tests during which
weight loss is recorded as non-evaporable water, the number in the parentheses is
heat rate in °C/min.
b
: The desiccant used was anhydrous Mg(ClO4)2, which became a mixture of di- and
tetra-hydrates after combining with all the water given up by the sample.
c
: Weight loss during 600-800°C was excluded (considered to be carbon dioxide).
a
14
1.2.3 Tensile strength tests
The traditional methods used to obtain the tensile strength of concrete materials include
direct (uniaxial) tension test, beam flexural test, and splitting tensile test. The last two test
methods are indirect in the sense that tensile stress is applied indirectly to the specimens. Tensile
strengths measured by indirect tests are inaccurate because they are evaluated based on the
assumption of linear elastic behavior. On the other hand, direct tension tests are difficult to
conduct without the grips of the testing device introducing eccentricity and secondary stresses.
The indirect methods are more widely used due to their simplicity. Results of flexural tests,
known as modulus of rupture, are usually more than 30% higher than those obtained in direct
tension tests (Neville 1996). Results of splitting tensile tests are reported to be -2 to 12% higher
than those obtained in direct tension tests, depending on specific test conditions (Ministry of
Transport of China 1981, CEB-FIB Code 1991, Neville 1996, Zheng 2001). However, Lin (2003)
determined by numerical analysis that the ratio of uniaxial tensile strength to splitting tensile
strength should be in the range between 1.09 and 1.41, depending on the tensile strength to
compressive strength ratio and the bearing strip width used in splitting tensile tests. Rocco (1999)
studied the effect of specimen diameter and load-bearing strip properties on splitting tensile
strength of mortar and granite and concluded that the size effect became negligible when
specimens were tested with bearing strips of small relative width (b/D < 0.08), which is defined
as the ratio between the bearing strip width (b) and the specimen diameter (D). For specimens of
the same size, the measured splitting tensile strength decreased with decreasing bearing strip
width and the effect was much more remarkable for smaller specimens. Differences of up to 30%
were found for specimens with a diameter of 37 mm when b/D was reduced from 0.16 to 0.08.
15
Fluid pressure testing of tensile strength of concrete materials has not been widely adopted
due to uncertainty of the fracture mechanism. Nevertheless, the tensile strength obtained from
fluid pressure tests can be very close to that obtained from traditional tests, especially when
nitrogen gas (whose viscosity is about two orders of magnitude lower than that of water) is used
as the pressurizing medium. Most of the earlier studies were performed on cylindrical specimens
with free ends, i.e. only annular pressure was applied. Clayton (1978) was among the first to
study the correlation between fluid pressure and splitting tensile test results. The correlation
factor (α), defined here as the ratio between fluid pressure tensile strength (ffpt) and splitting
tensile strength (fst), was found to decrease with decreasing loading rate, which ranged from 12
to 0.012 MPa/min. For nitrogen-pressure tests, α appeared to have reached a plateau (≈ 1) when
the loading rate is reduced to the range between 1.2 and 0.12 MPa/min. Similar correlation
factors (1.01-1.11) were obtained by Mindess (2005) using a loading rate of 0.6 MPa/min.
However, for water pressure (hydraulic fracture) tests, α continued to decrease (from 1.89 to 1.25)
with decreasing loading rate, suggesting that a plateau has not been reached. Recently, water
pressure tests on more permeable materials such as Leuders limestone showed that α might be
close to 1 at a loading rate of 0.69 MPa/min (100 psi/min) (Meadows 2009). Additionally, a few
tests were performed on cement paste specimens with both oil (which has a higher viscosity than
water) and water as the pressurizing medium and the fracture pressure of the former was found to
be much higher than that of the latter (Meadows 2009). Therefore, the fluid pressure tensile
strength appeared to decrease with decreasing loading rate until a threshold rate was reached,
below which test results were independent of loading rate and approximately equal to splitting
tensile strength. This threshold value could be increased (to avoid unusually long test duration)
by either decreasing fluid viscosity or increasing specimen permeability. For tests performed
16
with a fixed loading rate above the threshold value (such as in this study), the correlation factor α
would probably decrease with decreasing fluid viscosity and increasing specimen permeability.
1.3 Outline of the Dissertation
The dissertation consists of eight chapters.
Chapter 1 provides an introduction to Portland cement and some background information
for this study. A brief review of the commonly used methods to measure chemical shrinkage,
hydration kinetics, and tensile strengths of Portland cement pastes is also included.
Chapter 2 gives detailed descriptions of the experimental program. This includes
characterization of the test materials (cement), design and configuration of the test apparatus, as
well as development of test protocols. The entire test program was divided into three parts:
preliminary tests, pressure cell tests, and isothermal calorimetry tests. The preliminary tests were
conducted mainly to investigate the influencing factors of splitting tensile strength and its
relationship with direct tension strength. The pressure cell tests focused on the effects of curing
temperature and pressure on the various properties of different oilwell cements, including
chemical shrinkage, density, and tensile strength (measured by both water pressure and splitting
tensile tests). The pressure cell test data were also used to study the damage mechanism of the
specimens during depressurization. Isothermal calorimetry tests were used to study the heat
evolution of different oilwell cements at different curing temperatures.
Chapter 3 studies the correlations between chemical shrinkage, non-evaporable water
content, and the degree of hydration of cement by analyzing the properties of different types of
water in a hydrated cement paste based on previously published experimental data. As both
chemical shrinkage and non-evaporable water content are approximately equal to the degree of
17
hydration of cement when normalized, multi-linear empirical models are developed to estimate
the normalization factors (i.e. the total chemical shrinkage (CS0) and non-evaporable water
content (wn0) at complete hydration) based on cement composition. Since CS0 also depends on
curing conditions, methods of calibrating it for different curing temperatures and curing
pressures is also provided.
Chapter 4 presents the hydration kinetics data of different types of cement at different
curing conditions derived from the chemical shrinkage test results of this study using the model
developed in Chapter 3. A simple one-parameter model is proposed to reproduce the hydration
kinetics curves of cement at any curing temperature and pressure from the experimental data of a
reference condition using simple coordinate transformations (i.e. rescaling the x and/or y axis
using a single scale factor). The relationship between the scale factor and curing condition is
modeled by chemical kinetics theories.
Chapter 5 further investigates the hydration kinetics of cement, particularly that of its main
component, C3S. A particle based numerical model is developed to model C3S hydration in
stirred dilute suspensions and verified with experimental data published in the literature. After
some minor modifications, the model is then applied to experimental data of this study, i.e.
cement paste hydration during early ages (2-3 days). Since both C3S and C3A contribute
significantly to cement hydration at early stages, application of the model is currently limited to
those cements with no C3A content. Based on the fitted results of the model, a new explanation
of cement hydration kinetics is proposed.
Chapter 6 presents the hydration kinetics data of different types of cement at different
curing temperatures derived from the isothermal calorimetry test results of this study. The oneparameter model developed in Chapter 4 is further verified for the effect of curing temperature
18
with these test data. Some limitations of the model are also discussed. The agreements between
the test results of the two indirect methods of measuring cement hydration kinetics, namely
chemical shrinkage and isothermal calorimetry tests, are evaluated. The variations of their
correlation factors with curing temperature are investigated.
Chapter 7 presents the density and the tensile strength data of the cement pastes. The
influencing factors of the splitting tensile strength of cylindrical cement paste specimens are
discussed to determine the most appropriate way of conducting splitting tensile tests. The effects
of curing temperature and pressure on the density of the set cement (which is closely related to
mechanical properties) are studied in this chapter. Finally the effects of curing temperature and
pressure on the tensile strengths obtained by both splitting tensile tests and hydraulic fracture
tests are investigated. Potential damage to specimens caused by depressurization are evaluated
by observing the pattern of fracture planes. A damage mechanism is proposed after analyzing the
system deformation behavior during the depressurization process.
Chapter 8 summarizes the most important findings of this work. Main conclusions are
made and possible future works are proposed.
19
CHAPTER 2 : EXPERIMENTAL METHODS AND PROGRAM
2.1 Materials and Methods
All cements used in the study were provided by Halliburton. They were packed separately
in sealed plastic bags which were placed in sealed buckets for shipment and storage. A total of
four different classes of oilwell cements, namely Class A, C, G, and H were investigated. Two
different types of Class H cement were used: i.e. premium Class H (H-P) and standard Class H.
The standard Class H cement was provided in two batches (H-I and H-II), which were found to
have slightly different compositions. Cement oxide analysis results are presented in Table 2.1.
The main compound compositions of the different types of cement calculated according to API
Specification 10A (2010) are presented in Table 2.2 (CaO values used in the calculation were
corrected for free lime content). The particle size distributions (PSD) of the cements were
measured by laser scattering tests with dry dispersion methods. The average test results (at least
10 measurements were performed on each type of cement) are presented in Figure 2.1. The PSD
of Class H-I cement (not measured) should be similar to that of Class H-II since they are the
same type of cement from the same manufacturer. The median particle sizes for Class A, C, G,
H-P, and H-II cements were 38, 15, 34, 30, and 23 µm, respectively, while their specific surface
areas calculated from the PSD data (assuming spherical particles and a density of 3.15 cm3/g)
were 3562, 5649, 3265, 3939, and 3230 cm2/g, respectively. It is noticed that the particle size
distribution curves for Class A, G, and H-P cements are very similar, suggesting that probably
similar grinding procedures were adopted in manufacturing these cements. The specific surface
areas are similar for the different types of cement except for Class C, which was ground much
finer than the others to achieve a higher hydration rate.
20
Table 2.1: Oxide analysis results of the different types of cement
(in % after corrected for loss on ignition)
Cement
A
C
G
H-P
H-I
H-II
Na2O
0.15
0.26
0.19
0.13
0.12
0.12
MgO
0.98
2.51
0.72
1.04
2.55
2.43
Al2O3
5.13
3.29
4.09
3.39
2.92
2.69
SiO2
20.42 20.83 22.03 22.19 21.57 21.47
SO3
2.74
2.79
2.26
2.47
2.67
2.83
K2O
0.92
0.46
0.54
0.44
0.26
0.16
CaO
66.16 65.61 66.30 63.49 65.10 65.60
TiO2
0.25
0.23
0.17
0.19
0.24
0.25
MnO
0.05
0.03
0.06
0.10
0.05
0.06
Fe2O3
3.09
3.88
3.57
6.47
4.40
4.23
ZnO
0.02
0.03
0.01
0.02
0.04
0.07
SrO
0.08
0.08
0.06
0.07
0.09
0.09
LOI
2.38
1.49
0.58
1.03
0.62
0.83
Na2O and MgO values were determined by inductively
coupled plasma emission spectrometry (ICP) while all
other values were determined using X-ray fluorescence
spectroscopy (XRF).
Table 2.2: Estimated main compound compositions of the different types of cement
Cement
A
C
G
H-P
H-I
H-II
C3 S
61.66
72.24
62.62
47.91
66.52
70.32
C2 S
12.01
5.21
15.90
27.46
11.65
8.49
C3 A
8.36
2.16
4.80
0
0.29
0
C4AF C2F CaSO4 Free Lime
9.41
0
4.67
1.43
11.82
0
4.74
0.23
10.87
0
3.84
0.21
16.17 1.97 4.21
0.30
13.40
0
4.54
0.26
12.83 0.03 4.81
0.34
21
Figure 2.1: Particle size distributions of the different types of cement
All slurries (cement pastes) were prepared with deaerated water and cement only with no
additives. Deaerated water was prepared by boiling and sealing regular tap water in flasks.
Because relatively large batches are needed for the test (larger than the capacity of a typical
blender), slurries are prepared with a two-speed Waring laboratory blender and a three-speed
Hobart cement and mortar mixer, both of which were rinsed and wiped with a wet paper towel
before use. All slurries were prepared at ambient temperatures. The procedure to prepare a slurry
consisted of the following steps: 1) Weigh mixing water and cement in an appropriate number of
sets (typically three sets) according to the size of the batch such that the volume of each set does
not exceed the capacity of the blender. 2) Pour mixing water into the blender jar and turn the
blender (sealed with a cap) on at low speed. 3) Remove the plug in the center of the cap to
slowly introduce cement through a funnel. 4) After all cement is introduced, switch the blender
to high speed to mix for another 35 seconds and then transfer the slurry to the mixer. 5) Repeat
22
steps 2 through 4 for the remaining sets. 6) Blend the entire batch in the mixer at the lowest
speed for an additional 15 minutes (7 minutes for Series II of pressure cell tests, see Section
2.3.2). Please note that mixing in the Hobart mixer, which uses a wire whip at 60 rpm, is only
adopted to make sure the entire batch is thoroughly mixed and has very little effect on cement
hydration.
2.2 Test Apparatus and Procedure
Development and manufacturing of the device suitable for in-situ testing of oilwell cement
underwent several iterations, which lasted more than four years (Meadows 2009). A US patent
(Funkhouser 2009) was granted covering the basic design. The final product consists of four
identical steel pressure cells, each of which can be placed in a constraining steel frame to allow
inside pressure to build up safely. All cells are connected with water-filled injection pumps
through tubing and fittings. The system configurations and test procedures are slightly different
for the two series of pressure cell tests performed in this study, which will be discussed further in
Section 2.3. This section presents the most ideal test set up and test procedure suitable for in-situ
testing.
Figure 2.2 shows a sketch of the pressure cell and the reaction frame. The pressure cell
consists of a hollow steel cylinder and two end caps fitted with O-rings. As shown in the sketch,
the end caps of the pressure cell are covered with filter paper to prevent cement from entering the
tubing system. Cement specimens are cast inside a removable rubber sleeve, which has a
diameter of 50mm and a height of 170mm. The rubber sleeve is perforated in the region between
the annular seals to allow direct contact between water and the specimen for hydraulic fracture
23
testing. The perforations are also covered with filter paper to prevent cement leakage and to
allow even distribution of water pressure.
Figure 2.2: Sketch of the pressure cell (not to scale)
24
The specimens were found to have perfect bonding with the rubber in the regions not
covered by filter paper, especially for ambient-temperature tests. The “effective” specimen (i.e.
the section covered by filter paper and subjected to hydraulic fracture) has a diameter of 50mm
and a height of 75mm. The total height of the rubber sleeve is 170mm. The extra heights of the
specimen are used to make sure that there is adequate isolation between the annulus region and
the end regions. The final height of hardened cement specimen is usually shorter than that of the
rubber sleeve due to settling and bleeding (free water). Combination of high temperature (≥
40 °C) and high pressure (≥ 6.9 MPa) curing can cause the rubber sleeve to “buckle” in the
region just above the top of the specimen, which sometimes results in cracking. The problem can
be solved by cutting the rubber sleeve shorter and attaching a piece of protruding filter paper at
the top to make sure the cement specimen is higher than the rubber sleeve even after settling.
Note that such treatment of the rubber sleeves is not necessary for tests conducted at ambient
temperatures or at the relatively low curing pressure of 0.69 MPa.
Pressure control of the system is achieved by three injection pumps manufactured by
Teledyne ISCO. Figure 2.3 shows the configuration of the entire system. Pumps A, B, and C
have volume capacities of 260, 1000, and 1000 mL, and pressure ratings of 51.7, 13.8, and 13.8
MPa, (7500, 2000, and 2000 psi) respectively. Pump A pressurizes the annulus surfaces of the
specimens. Pump B pressurizes the circular end surfaces at the top and bottom of the specimens.
Pump C applies pressure to the active seals which isolate the end from the annular regions of the
specimens. To assure such isolation, the active seal pressure needs to be greater than both the
pressures applied to the annular and end regions. Three valves are used to control the annular,
end, and seal pressures of each cell, respectively. The annular and end regions are interconnected
25
through Isolation Valve #1 while the end and seal regions are interconnected through Isolation
Valve #2. When both isolation valves are open, a uniform hydrostatic pressure can be applied to
all four specimens by any connected pump for curing the cement and measuring chemical
shrinkage, which can be obtained by simply recording the total volume of water entering the cell
(i.e. the volume change of the injection pump). In this study, Pump A was used for chemical
shrinkage measurements due to its better leak tightness (valves to Pumps B and C are closed
during the curing period). Temperature control of the system is achieved by four heat controllers
independently connected to the heating tapes wrapped around the cells. Heat insulation was
provided by enclosing the cells with two pieces of fiberglass insulation sheet. Thermocouples
were attached to the outside of the cells to prevent overheating. The temperature controller has a
hysteresis of 2.8 °C (5 °F).
Figure 2.3: Schematic of the test system
26
The procedures of preparing a test with the pressure cells are as follows: 1) Discharge all
the water in Pump A and fully refill it at a rate of 20 mL/min with deaerated water. 2) Assemble
the connections according to Figure 2.3, except the top caps of the pressure cells. 3) By properly
controlling the pumps, the valves, and the connections, purge the air in the tubing system as well
as in the conduits inside the pressure cells; a small amount of water typically accumulates at the
bottom of the pressure cells after this. 4) Place a piece of filter paper covering the bottom cap of
each pressure cell, which absorbs part of the accumulated water; suck up the excessive water
with paper towels. 5) Put a piece of filter paper inside each rubber sleeve covering the
perforations, wet the filter paper by dipping it in water and dry the uncovered sections of the
rubber sleeve. 6) Put the rubber sleeves in the pressure cells. 7) After the slurry is prepared as
described in Section 2.1, pour it into a beaker and record the total weight of the cement slurry,
the beaker, and a steel rod. 8) Pour the slurry into a pressure cell till it is half full. 9) Puddle the
slurry about 27 times with the steel rod to remove trapped air and fill voids. 10) Fill the pressure
cell till the top of the rubber sleeve and repeat Step 8. 11) Record the remaining total weight of
the slurry, the beaker, and the rod such that the initial liquid weight of a specimen cast in the
pressure cell can be derived. 12) Repeat Step 7-11 for the other three pressure cells. 13) Put a
piece of filter paper on top of each specimen in the pressure cell. 14) Fill the pressure cell with
deaerated water till near completely full. 15) Put on the top caps and squeezing out excessive
water through the ports in the caps. 16) Slide the pressure cells into the constraining frames and
connect the top caps to the tubing system. 17) Open all the valves except those connected to
Pumps B and C. 18) Run Pump A at a constant flow rate of 5 mL/min to compress the system
until about 0.69 MPa. 19) For higher pressure tests, increase the curing pressure to the target
27
value at a constant rate of 3.45 MPa/min (500 psi/min). 20) For temperature controlled tests, set
the initial temperature 8.3 °C (15 °F) above the target value and hold for about 10 minutes after
the set temperature is reached, before resetting it back to the target value.
Since the temperature control is achieved through thermocouples attached to the outside
of the pressure cells, it is necessary to investigate the correlations between the temperature of the
pressure cells and that of the cement specimen. A few tests were performed by manually
recording the outside temperature of the pressure cells (i.e. readings on the temperature
controller) and the center temperature of the cement specimen (measured by a digital multimeter)
at an approximate time interval of 30 seconds. Unfortunately, such tests can only be performed
without closing the top cap of the pressure cell and without heat insulation. Figure 2.4 shows the
test results at 40.6 °C (105 °F). Note that the target temperature set with the heat controller needs
to be about 1.7 °C (3 °F) higher than the desired slurry temperature. It is obvious that the
temperature control scheme succeeded in quickly raising and stabilizing the slurry temperature.
The center temperature of the specimen is found to fluctuate much less than the outside
temperature of the pressure cell, which is associated with the hysteresis of the heat controller.
However, test results appear to be not very consistent. A better temperature control shall be
sought in the future.
28
50
50
Pressure cell
Specimen
Target
45
Temperature (°C)
Temperature (°C)
45
40
35
30
40
35
0
0.5
1
1.5
Time (h)
2
2.5
30
8.8
9.8
Time (h)
Figure 2.4: Temperature evolutions of the pressure cell and of the specimen
The tensile strengths of the specimens are determined with the hydraulic fracture method
after a specific age is reached. When the specimen in a particular cell is being tested, valves to all
other cells are closed. In a typical test, both isolation valves are closed and pressure is first
applied to the active seals by Pump C such that pressures of the end and annulus regions can be
controlled independently. Three different testing procedures can be used for fluid pressure
testing: In Procedure A, the pressure applied to the annular surface of the specimen is increased
at a constant rate while that applied to the ends is kept constant; in Procedure B, the pressure
applied to the ends of the specimen is decreased at a constant rate while that applied to the
annular surface is kept constant; in Procedure C, the pressure applied to the annular surface of
the specimen is increased while that applied to the ends is decreased simultaneously, both at
constant rates. The pressure difference between the end and the annulus regions when the
specimen fractures is assumed to be the water pressure tensile strength of the specimen. After the
hydraulic fracture tests, the specimens are removed from the pressure cells and cleaned to record
29
their final weights. A 50 mm high cylinder can be cut from the middle section (covered by filter
paper) of each specimen for splitting tests. Note that for pressure cell specimens all splitting tests
were performed at ambient temperatures using basswood bearing strips with a width of about 4
mm (resulting in a b/D ratio of 0.08) and a thickness of about 1 mm. The specimens, especially
those cured at high temperatures, shall be kept saturated (submerged in water) during preparation
to prevent premature damage due to excessive drying.
2.3 Test Program
2.3.1 Preliminary tests
The preliminary series of tests involved specimens cast in regular molds only. As
discussed in Section 1.3.3, the splitting tensile strength varied considerably with specific testing
parameters, which made it difficult to establish its correlation with results of other test methods.
A total of five batches of briquette specimens and cylinder specimens with varied height were
produced to study the two major influencing factors of splitting tensile strength: specimen
position and bearing strip width. All specimens were cured at atmospheric pressure and ambient
temperature. Direct tension tests were performed on briquette specimens while splitting tests
were performed on cylinders (51mm diameter and 51mm height) cut from the specimens. Two,
three, and five disks were cut from cylinder specimens cast with the heights of 102, 178, and
305mm, respectively (Figure 2.5). The width of the bearing strips used for splitting tests varied
from 2 to 25 mm, resulting in the relative width (b/D) varying from 0.04 to 0.5. A relative width
of 0 means no bearing strip was used between specimen and loading platen. The detailed test
scheme is listed Table 2.3.
30
Table 2.3: List of number of specimens tested at different conditions
No. of cylinder specimens
No. of
Cylinder
briquette
Relative width (b/D)
size
Total
specimens
0.5 0.16 0.08 0.04 0
PL-1
3
3
1
4
51x178
0
PL-2
3
6
2
8
51x178
6
a
PL-3
3
3
3
6
51x178
6
PL-4
2
4
4
8
51x305
6
PL-5
3
16
16
51x102
6
a
: The b/D ratio was later found to be around 0.09 due to imprecise cutting.
Test
No.
Test age
(days)
Figure 2.5: Tested specimens cut from cylinders of different heights
(178mm front, 305mm back)
2.3.2 Pressure cell tests
Two main series of tests were performed with the pressure cells. In the first test series, only
the standard Class H cement (H-I and H-II) with a w/c ratio of 0.4 was used. Due to the pressure
limits of the pumps, specimens cured at 0.69 MPa could only be tested by increasing the annulus
pressure (i.e. Procedure A as discussed in Section 2.2) while those cured at 13.1 MPa could only
be tested by decreasing the end pressure (Procedure B). Specimens cured at 6.9 MPa were tested
31
with both procedures, A and B. This resulted in a total of four testing schemes for each
temperature. Table 2.4 shows all 12 possible combinations of curing conditions and testing
schemes. Class H-I cement was only tested at ambient temperatures while Class H-II cement was
tested at all temperatures. The ambient temperatures measured by periodically recording the
temperature of the pressure cells was approximately 24 °C, whose long-term fluctuations (i.e.
temperature differences between different tests) may have been as high as ±2.8 °C, while those
short-term fluctuations (i.e. for a single test) were typically less than ±1.1 °C. Temperature
control with the heat controllers was only adopted for high temperature tests. The system
configuration for this test series was slightly different from Figure 2.3 in that the end and seal
regions were not interconnected. Such configuration has very little effect on test results but
sometimes resulted in damages to the active seals when seal pressure was not applied during the
curing period. The procedures of producing specimens for high temperature tests were also
slightly different from those described in Section 2.2 in that the pressure cells were preheated
before the slurry was introduced to simulate the field conditions. In this test series, all specimens
were tested in-situ using water pressure at a loading rate of 0.69 MPa/min after being cured for
48 hours. Hydraulic pressure was immediately released after the hydraulic fracture tests.
Table 2.4: Pressure cell tests (Series I, w/c = 0.4, Test age = 48 hours)
Curing Pressure (MPa)
Curing
Temperature
0.69
6.9
6.9
13.1
(°C)
Procedure A
Procedure B
a
c
Ambient
24-I (7) 24-II (4) 24-III (5) 24-IV (2)
40.6b
40-I (1) 40-II (1) 40-III (1) 40-IV (3)
b
60
60-I (2) 60-II (2) 60-III (2) 60-IV (2)
a
: Lab temperature (~24±2.8 °C)
b
: Estimated cement specimen temperature, about 1.7 °C lower
than the target value set on the heat controller.
c
: Number in parentheses indicate the number of batches tested.
32
In the second test series, all the different types of cement described in Section 2.1 were
investigated. Standard w/c ratios for each class of cement were used, as defined in API
Specification 10A (2010). The pressure range studied was nearly quadrupled compared with Test
Series I, because the latter showed that the effect of curing pressure on hydration kinetics were
too small to be effectively quantified. Table 2.5 shows the complete test scheme for this test
series. As seen in the table, two additional w/c ratios were used for Class H-II cement to study
the effect of w/c ratio on test results. Due to the pressure limits of Pumps B and C, it was not
possible to perform in-situ tests on specimens cured at high pressures (≥ 17.2 MPa). Therefore,
for the purpose of consistency within this test series, all specimens were depressurized (to 0.14
MPa) at a rate of 0.345 MPa/min before hydraulic fracture tests. Hydraulic fracture tests were
conducted at a loading rate of 0.69 MPa/min using Procedure A at the age of 72 hours.
Table 2.5: Pressure cell tests (Series II, Test age = 72 hours)
Curing Temperature
60b
Ambienta
40.6b
(°C)
psi
100
2500
5000
7500
100
100
Curing
Pressure
MPa
0.69
17.2
34.5
51.7
0.69
0.69
Cement
w/c
A
0.46c
A-1
A-2
A-3
A-4
A-5
A-6
C
0.56c
C-1
C-2
C-3
C-4
C-5
C-6
G
0.44c
G-1
G-2
G-3
G-4
G-5
G-6
H-P
0.38c
H-P-1
H-P-2
H-P-3
H-P-4 H-P-5 H-P-6
c
H-I
0.38
H-I-1
H-I-2
H-I-3
H-I-4
H-II
0.38c
H-II-1
H-II-2
H-II-3
H-II-4
H-II
0.3
H-II-3-1 H-II-3-2 H-II-3-3
H-II
0.5
H-II-5-1 H-II-5-2 H-II-5-3
a
: lab temperature (~24±2.8 °C), see Table 4.2
b
: Estimated cement specimen temperature, about 1.7 °C lower than the target value
set on the heat controller.
c
: Standard w/c ratios in API Specification 10A (2010)
33
2.3.3 Isothermal calorimetry tests
Isothermal calorimetry tests were performed at three different temperatures for five
different types of cement to measure the heat of hydration (Table 2.6). The tests were performed
by Dr. Dale Bentz at the National Institute of Standards and Technology using a TAM Air
calorimeter. Slurry preparation procedures were similar to those described in Section 2.1 using
the same blender. However, it was not necessary to use the cement and mortar mixer since only a
small quantity of material (between 4.35 g and 5.09 g) was needed for each test. Two slurries
were prepared for each type of cement. One slurry was sampled for tests at 25 and 40 °C while
the other slurry was sampled for tests at 25 and 60 °C. The two samples taken from different
slurries and tested at the same temperature of 25 °C can be used to provide an indication of
variability in test results.
Table 2.6: Isothermal calorimetry tests (Atmospheric pressure, Test age = 168 hours)
Curing Temperature (°C)
Cement
w/c
A
0.46
C
0.56
G
0.44
H-P
0.38
H-I
0.38
25
A-25
C-25
G-25
H-P-25
H-I-25
40
A-40
C-40
G-40
H-P-40
H-I-40
60
A-60
C-60
G-60
H-P-60
H-I-60
2.4 Test Data Collection and Analysis
2.4.1 Test data collection and processing
Chemical shrinkage was measured by recording the volume change of the syringe pump
connected to the pressure cells. The tests were timed when cement first came into contact with
water. Data collection rate was once per 0.5 second during the pressure application process (test
34
data were used for other studies), and was changed to once per minute as soon as the temperature
and pressure were stabilized. However, as it typically took more than half an hour for slurry
preparation and for temperature and pressure to stabilize, total chemical shrinkage was calculated
with the 0 point set at 1 hour for the purpose of consistency. Unlike total chemical shrinkage, the
rate of chemical shrinkage is not affected by the initial value and hence could be calculated as
soon as temperature and pressure were stabilized. In order to reduce data oscillation, the rates
were calculated at approximately equal intervals of total chemical shrinkage (about 1/300 of the
final value) for tests conducted at ambient temperatures. However, for tests performed at high
temperatures, data oscillations (caused by the hysteresis of the heat controllers) were too large to
allow direct derivation of accurate derivative curves. In these cases, it was found that averaging
test data of repeated tests (which are highly reproducible, Figure 2.6) can help to somewhat
reduce the oscillations. When the averaged test results are divided into three different sections,
each section can be smoothed by fitting with a sixth order polynomial function with fairly good
accuracy. Relatively reliable derivative curves can be obtained by differentiating these functions.
Figure 2.7 and Figure 2.8 show examples of test results before and after being smoothed. The
three separate sections typically correspond to the acceleration, the deceleration, and the steady
state period of hydration, respectively. However, there are no exact boundaries between these
different periods. The best fit functions were usually obtained by trial and error (which can be
easily achieved using a computer program).
35
0.03
0.025
3.5
3
0.02
2.5
0.015
2
1.5
0.01
1
Chemical Shrinkage (%)
Chemical Shrinkage (ml/g cement)
4
0.005
0.5
0
0
10
20
30
Time (h)
0
50
40
Figure 2.6: Four repeated tests at curing temperature of 60 °C and curing pressure of 6.9 Mpa
(i.e. Tests 60-II and 60-III)
3
0.03
2.5
0.025
2
0.02
Before smoothing
After smoothing
1.5
1
0.01
0.5
0
0.015
0.005
0
10
20
30
Time (h)
40
Total chemical shrinkage (mL/g cement)
Rate of chemical shrinkage (mL/h/g cement)
-3
x 10
0
50
Figure 2.7: Average test results of Test 40-IV before and after smoothing
36
7
0.028
6
0.024
5
0.02
4
0.016
Before smoothing
After smoothing
3
0.012
2
0.008
1
0.004
0
0
10
20
Time (h)
30
40
Total chemical shrinkage (mL/g cement)
Rate of chemical shrinkage (mL/h/g cement)
-3
x 10
0
Figure 2.8: Average test results of Test 60-IV before and after smoothing
When the specimens had reached the specified age, the data collection program was
restarted to record hydraulic fracture test data from all three pumps at a rate of once per 0.5
second. In a hydraulic fracture test, the pressure difference between the annulus and the ends of a
specimen was typically programmed to increase linearly with time (at a rate of 0.69 MPa/min)
until the specimen fractured. It was found that hydraulic fracture cannot be achieved if there is
no direct contact between water and the specimen (i.e. when impermeable rubber sleeves are
used). Figure 2.9 shows the pressure vs. time plot of a specimen tested with procedure A. The
flow rates that the pumps have to maintain in order to keep the pressures as programmed are also
shown. A flow dynamic clearly exists within the specimen as the pump flow suggests that water
flows from the high pressure region (annulus) to the low pressure regions (the ends). The flow
rate, which is shown to gradually increase with time and the pressure difference, experiences an
abrupt increase just as the specimen fractures. However, occasionally the fracture point was not
so easily identified, especially in tests where Procedure B was used. In these cases, the specimen
37
was assumed to have fractured at the point of an apparent change of slope of the pressure-time
curve, which was usually accompanied by a change of slope of the flow rate-time curve. It
should also be noted that the flow rate might vary significantly from one specimen to another and
did not always exhibit a linear relationship with the pressure difference. The existence of a flow
dynamic during a test usually resulted in a gradual and slight pressure deviation in the region
where pressure was supposed to be kept constant (i.e. the end regions for procedure A and the
annulus region for procedure B). Such deviation was normally less than 0.07 MPa, but could be
much higher when the flow rate was unusually high, possibly due to inadequate isolation
between the annulus and the end regions. Samples were normally produced in sets of four.
However, the number of valid tests was sometimes less than four in one set due to equipment or
operator errors. Chauvenet’s criterion (Taylor 1997b) was used to identify potentially spurious
data points (i.e. outliers), which were excluded from analysis.
Pressure (MPa)
4
Annulus Pressure
End Pressure
Pressure Diff.
3
2
1
0
0
0.5
1
1.5
2
2.5
Time (min)
3
3.5
4
4.5
1.5
2
2.5
Time (min)
3
3.5
4
4.5
Flow rate (mL/min)
2
Annulus
End
1
0
-1
-2
0
0.5
1
Figure 2.9: Test plot of a specimen cured at 0.69 Mpa and 24 °C
38
2.4.2 Influence factors of chemical shrinkage test results
2.4.2.1 Effect of system deformations
Since chemical shrinkage is measured by monitoring the deformation of the entire system
inside the test apparatus, the accuracy of test results can be evaluated by estimating the system
deformation caused by factors other than chemical shrinkage. The procedures of performing
system deformation tests are similar to standard chemical shrinkage tests except that all pressure
cells are filled with deaerated water instead of cement slurries. Test results obtained at different
pressures are shown in Figure 2.10. It is observed that both the total deformation and the
fluctuations of test results during the 72-hour period decrease with increasing pressure. The
majority of the total deformation occurred during the first 10 to 20 hours. There is not yet a clear
explanation of these deformation behaviors. Possible contributing factors include: compression
and dissolution of entrapped air, leakage, hysteresis of the pressure control system, and plastic
deformation of the rubber sleeves and the test apparatus. The entrapped air, which seems to be
unavoidable during pump refilling and test preparation, is only likely to affect tests conducted at
0.69 MPa because with increasing pressure its volume decreases significantly and its solubility in
water increases significantly. This explains why the deformation measured during the postpressurization (constant pressure) period is less for higher pressures. In addition, test results at
the relative low pressure of 0.69 MPa seem to be inconsistent, which may be attributed to the
different amount of entrapped air. Figure 2.10 suggests that chemical shrinkage test results
obtained at high curing pressure are probably more reliable. Considering that the total chemical
shrinkage of different cements at the age of 72 hours is generally larger than 50mL, the
maximum error associated with system deformations is approximately 3%.
39
0
Pump volume change (mL)
34.5 MPa
-0.5
17.2 MPa
-1
0.69 MPa
-1.5
-2
0
10
20
30
40
Time (h)
50
60
70
80
Figure 2.10: System deformation tests performed at different pressures
In order to further study the deformation behavior of water under pressure, a few
pressurization and depressurization tests with constant pressure gradients were performed with
different amounts of water using different systems. Some tests were performed with the syringe
pump only by closing both its inlet and outlet valves while others were performed with the entire
test system (i.e. Pump A and four pressure cells). Test results were found to be largely
independent of pressure gradients, which ranged from 0.345 to 3.45 MPa/min. There was also
very little difference between the loading and the unloading plots of pressure vs. volume. The
apparent bulk modulus of water can be calculated from these tests according to the following
equation,
K = −V
∂P
∂V
(2.1)
where K is the bulk modulus, V is the volume, and P is the pressure. Some representative test
results are shown in Figure 2.11. Due to difficulties associated with estimating the “dead volume”
40
within the system (i.e. volume of water in the valves, tubing, and ports, etc.), it seems very
difficult to obtain accurate values of bulk modulus of water with these test systems. Theoretically,
the bulk modulus of water should increase steadily with increasing pressure (NIST 2011). But all
test results in this study show an abrupt change in the pressure range around 2 to 3 MPa. The
relatively high compressibility (low bulk modulus) measured at low pressures is probably caused
by compression and dissolution of entrapped air. Although the amount of entrapped air may vary
from one test to another, Figure 2.11 suggests that a pressure of 3 MPa or higher is generally
required to minimize the effect of entrapped air on test results. Hence, the chemical shrinkage
test results obtained at the relatively low pressure of 0.69 MPa in this study may be less accurate
than those obtained at higher pressures. This conclusion is consistent with system deformation
test results shown in Figure 2.10.
2
Bulk modulus (GPa)
1.8
1.6
1.4
Pump A (260 mL)
Pump B (1000 mL)
Pump A + Cells (1800 mL)
1.2
1
0
5
10
Pressure (MPa)
15
Figure 2.11: Apparent bulk modulus of water at different pressures
41
2.4.2.2 Effect of temperature fluctuations
The main shortcoming of the newly developed apparatus is the lack of a precise
temperature control scheme. As shown earlier, when the heat controllers were used, test results
became oscillatory due to the hysteresis. When they were not used, test results were influenced
by temperature fluctuations in the lab. The effect of the heat of hydration on the temperature of
the specimens seems to be very small probably due to the fact that hydration heat is readily
absorbed and dissipated by the pressure cells, which possess a relatively large thermal mass. As
discussed earlier, the lab temperature was usually fairly constant during the duration of a single
test, but might fluctuate to some extent over longer periods. Therefore, tests conducted within a
short time span (Figure 2.12) show considerably smaller variations compared with those
performed relatively far apart from each other (Figure 2.13). The variation in peak hydration rate
obtained from different tests was about 2% for the former and about 7% for the latter. The effect
of temperature fluctuations on total chemical shrinkage at the end of 48 hours appears to be very
small (approximately 2%) in both cases. It should also be noted that test results at the relatively
low curing pressure of 0.69 MPa are naturally less accurate than those of the other tests due to
the effect of entrapped air discussed in the previous section. As indicated in Figure 2.12, one test
performed with impermeable rubber sleeves (with no filter paper) generated the same results as
the standard tests. Since there is perfect bonding between cement and the rubber sleeve, the
results suggest that specimen thickness (up to 170 mm) is not a limiting factor of chemical
shrinkage test results for the cement slurry studied here. It is also noticed that the derivative
curves of Class H-I cement are distinctively different from those of Class H-II cement (with a
small bump just after the main peak) due to the small amount of C3A content.
42
-3
0.03
Standard tests
Impermeable rubber sleeve test
1
0.02
0.5
0.01
0
0
10
20
30
40
Total chemical shrinkage (mL/g cement)
Rate of chemical shrinkage (mL/h/g cement)
1.5
x 10
0
50
Time (h)
Figure 2.12: Four repeated tests of 24-II/24-III (one with impermeable rubber sleeves)
(Class H-I cement, w/c=0.4)
-3
0.03
1
0.02
0.5
0.01
0
0
10
20
30
40
Total chemical shrinkage (mL/g cement)
Rate of chemical shrinkage (mL/h/g cement)
1.5
x 10
0
50
Time (h)
Figure 2.13: Four repeated tests of 24-I (Class H-II cement, w/c=0.4)
43
2.4.2.3 Effect of specimen thickness
Since the specimens used in this study are significantly larger than those in traditional
chemical shrinkage tests, it is important to investigate whether test results are affected by
specimen size. Specimen thickness is usually believed to be one of the most important limiting
factors of chemical shrinkage measurement. Therefore, ASTM (2007) specifies that the
specimen thickness should be between 5 and 10 mm. However, as discussed in Section 1.3.1,
Costoya (2008) has found that specimen thickness is not a limiting factor for Alite paste with a
w/c ratio of 0.4 and adopted a thickness of 17 mm for measuring chemical shrinkage. In this
study, two supplementary tests were performed with Class A and Class H-P cements after the
main test series to study the effect of specimen thickness on test results. The procedures of
preparing these tests are the same as described in Section 2.2, with the following two exceptions:
1) One solid steel bar was placed in the center of each pressure cell (secured with a ring-shaped
felt at the bottom) to produce hollow cylinder specimens. 2) The entire interior surfaces of all
rubber sleeves were covered with filter paper to prevent bonding and provide access to curing
water. The supplementary tests were conducted at a curing pressure of 0.69 MPa and at ambient
temperatures using the same w/c ratios as listed in Table 2.5. The hollow cylinder specimens had
a wall thickness of approximately 10 mm.
Test results of thin specimens (hollow cylinders with a wall thickness of 10 mm) are
compared with those of the thicker ones (solid cylinders with a 25 mm radius) in Figure 2.14.
Note that the hollow cylinders had full access to water over the entire length of its annular
surface while the solid cylinders only had partial access to water (i.e. the section covered by filter
paper). Both types of specimens had full access to water around their end surfaces. The observed
differences in test results of different types of specimens are mainly attributed to the slightly
44
different lab temperatures. The effect of different temperatures can be corrected using the model
developed in Chapter 4. Figure 2.15 shows the test results calibrated to a uniform temperature of
25.6 °C (78 °F). It appears that specimen thickness has virtually no effect on chemical shrinkage
test results, consistent with test results shown in Figure 2.12. Since the w/c ratios used in these
studies are relatively high, slurries with lower w/c ratios shall be further investigated in the future.
Total chemical shrinkage (mL/g cement)
0.04
0.035
0.03
~24.4 °C
~25.6 °C
0.025
~26.7 °C
0.02
~25.6 °C
0.015
Class
Class
Class
Class
0.01
0.005
0
0
10
20
30
40
Time (h)
50
A (solid)
A (hollow)
H-P (solid)
H-P (hollow)
60
70
Figure 2.14: Effect of specimen thickness on test results
80
45
Chemical shrinkage (mL/g cement)
0.04
Class A cement
0.035
0.03
0.025
0.02
Premium Class H cement
0.015
0.01
Solid cylinder specimen
Hollow cylinder specimen
0.005
0
0
10
20
30
40
50
Time (h)
60
70
80
Figure 2.15: Effect of specimen thickness on test results
(Calibrated to a uniform temperature of 25.6 °C)
2.4.4 Reproducibility of isothermal calorimetry tests
Compared with chemical shrinkage tests, the isothermal calorimetry test is a relatively well
established method of evaluating cement hydration kinetics. Many different types of commercial
calorimeters are available. The standard practice is described in ASTM C1679 (2009). Figure
2.16 shows the repeated test results of different cements obtained at a uniform temperature of
25 °C. Note that the test results of Class G cement, which almost overlap with those of Class A,
are not shown for the clarity of the Figure. As observed in the figure, excellent reproducibility is
observed for the two data sets (samples are taken from different slurries) obtained in this study.
Cumulative heat evolution (J/g cement)
46
400
350
300
250
200
Class C
150
Class A
100
Class H-I
50
0
Test Set 1
Test Set 2
Class H-P
0
50
100
150
Time (h)
Figure 2.16: Repeated isothermal calorimetry tests at 25 °C
47
CHAPTER 3 : CORRELATION BETWEEN CHEMICAL
SHRINKAGE AND THE DEGREE OF HYDRATION OF
CEMENT
3.1 Introduction
As shown in Eq. (1.2), the degree of hydration of cement is approximately equal to the
normalized chemical shrinkage. Therefore, their correlation with each other is determined by the
normalization factor CS0. Due to the multiphase nature of Portland cement, this normalization
factor (i.e. the total chemical shrinkage at the complete hydration condition) mainly depends on
cement compound composition and can be modeled by the following equation,
CS 0 = a1 ⋅ pC3S + a2 ⋅ pC2 S + a3 ⋅ pC3 A + a4 ⋅ pC4 AF
(3.1)
where pi is the Bogue weight fraction as defined in Section 1.2.2, while ai is a constant that
equals the total chemical shrinkage from the complete hydration of 1 g of the i-th compound in
cement (1 for C3S, 2 for C2S, 3 for C3A, and 4 for C4AF).
Theoretically, the coefficients (ai) associated with different clinker phases can be calculated
based on the hydration reaction stoichiometries. An example of chemical shrinkage calculated
for C3S is as follows:
Molar Mass (g/mol)
Density (g/cm3)
Mass (g)
Volume (cm3)
C3S + 5.3H → C1.7SH4 + 1.3CH
Chemical
228.3
95.5
227.5
96.3
Shrinkage
3.15
0.998
2.01
2.242
(mL/g C3S)
1
0.4182
0.9963
0.4219
0.3175
0.4190
0.4957
0.1882
-0.0526
Apparently, the calculation involves knowing the accurate chemical formulae and densities of all
reactants and resultants. While there is relatively good agreement with regard to the chemical
48
formulae and densities of water, gypsum, calcium hydroxide and all clinker phases (Taylor
1997a, Tennis 2000, Mounanga 2004, Bentz 1995), these properties of most hydration products
depend on their water content, which in turn depends on their drying condition (or the ambient
humidity). Since chemical shrinkage is measured with samples submerged in water, it is most
appropriate to use the chemical equations for reactions at saturated states. The hydration products
of C3S and C2S are C-S-H gel and calcium hydroxide. However, the exact chemical formula and
density of saturated C-S-H gel are still uncertain (the values used in the above example are only
approximate). In fact, it is highly likely that the densities of C-S-H gel formed at different stages
of hydration are different (Tennis 2000, Garrault 2006, Bishnoi 2009a). In addition, there are
many other possible hydration products for C3A and C4AF, whose exact densities are also
uncertain. Therefore, it is very difficult to accurately estimate the coefficients for CS0 using
reaction stoichiometries. Table 3.1 gives a summary of calculated results with properties of
hydration product from different sources. As shown in the table, a slight variation in density can
cause a significant variation in calculated chemical shrinkage.
The uncertainties associated with hydration reaction stoichiometries call for an alternative
method to estimate the coefficients (ai) for different phases. When a set of experimental CS0 data
are obtained for different cements with different compositions, the ai’s may be determined
empirically by performing a multi-linear least-square regression analysis according to Eq. (3.1).
Unfortunately, it is nearly impossible to directly measure total chemical shrinkage at later ages
due to reductions in the permeability of the samples (depercolation of capillary porosity) (Ye
2005, Bentz 2006, Sant 2009). CS0 can be estimated experimentally by extrapolating early age
test data of CS(t) as a function of α(t) (measured by QXRD) according to Eq. (1.2). However,
experimental data in the literature are not sufficient for multi-linear regression analysis due to the
49
fact that QXRD is rarely used to monitor the degree of hydration of cement. The non-evaporable
water content of a hydrated cement sample is much more easily obtained at later ages and has
been investigated extensively in many studies. Since chemical shrinkage, non-evaporable water
content, and the degree of hydration of cement are all closely related to each other, studying the
properties of different types of water in a hydrated cement paste can help to establish a relatively
reliable equation to estimate CS0.
Table 3.1: Total chemical shrinkage at complete hydration of different clinker phases
Clinker phase
Chemical
formula
Density
(kg/m3)
Hydration product
Chemical
formula
Density
(kg/m3)
Reference
Chemical
shrinkage
(mL/g)
1990a
Tennis 2000
0.048
Mounanga 2004
2010
0.053
C3 S
3150
C1.7SH4
b
2063
Costoya 2008
0.065
2120
Bentz 1995
0.078
a
1990
Tennis 2000
0.034
Mounanga 2004
0.041
2010
C2 S
3280
C1.7SH4
b
2063
Costoya 2008
0.058
2120
Bentz 1995
0.075
1700
Bentz 1995
0.159
1750
Tennis 2000
0.237
C6 AS 3 H 32
1775
Brouwers 2005
0.274
Mounanga 2004
1780
0.281
C6 AS 3 H 36
1720
Brouwers 2005
0.303
1990
Tennis 2000
0.115
C3 A
3030
C4 ASH12
2014
Brouwers 2005
0.129
Mounanga 2004
2020
0.132
C4 ASH14
Brouwers 2005
0.190
2003
2527
Brouwers 2005
0.177
C3AH6
2670
Tennis 2000
0.207
C4AH13
2046
Brouwers 2005
0.240
C4AF
3730
C3(A,F)H6
2670
Tennis 2000
0.148
a
: The optimised value from chemical shrinkage results
b
: Optimised value from chemical shrinkage results (average of 4 samples)
50
3.2 Classification of Water in Cement Paste
Despite the fact that the hydration reactions of different phases in Portland cement are
different, the causes of chemical shrinkage may all be attributed to the change of the state of
water (from free water to chemically combined and/or physically adsorbed water). Powers and
Brownyard (1946) classified the total water content in a cement paste into three categories:
capillary water (free water), adsorbed water (water bound by surface forces), and water of
constitution (chemically combined water). The authors also pointed out that such classification is
of little practical use, because it is impossible to experimentally separate the total water content
into such divisions. Therefore, an experimentally obtainable classification based on the volatility
of water was proposed: capillary water (the water lost when the relative humidity is decreased to
45%), gel water (the water lost when the relative humidity is decreased from 45% to near zero),
and non-evaporable water (the water lost when a dried cement paste is ignited at about 1000 °C).
This classification roughly corresponds with the three previous categories. Thus the nonevaporable water content in a sample is usually considered approximately equal to its chemically
combined water content.
3.3 Mass Fractions of Different Types of Water in Cement Paste
When cement is first mixed with water, all water can be classified as capillary water. For a
cement paste with a relatively high w/c cured under sealed or saturated conditions, its total water
content may decrease initially as a result of bleeding and increase subsequently due to water
imbibitions. With the progress of hydration, the mass fraction of capillary water reduces while
those of the non-evaporable water and gel water increase. Hydration stops when capillary water
is completely consumed or when cement is completely hydrated, whichever happens earlier.
51
Powers and Brownyard (1946) found that the mass fraction of gel water is approximately
proportional to that of non-evaporable water, with a proportionality constant ranging from 0.96
to 1.12 (gel water/non-evaporable water) depending on the type of cement. The non-evaporable
water content (wn0) of a completely hydrated sample should depend only on the cement
composition when expressed in terms of g/g of original cement. Similar to Eq. (3.1), the
following equation may be used to estimate wn0,
wn 0 = b1 ⋅ pC3S + b2 ⋅ pC2 S + b3 ⋅ pC3 A + b4 ⋅ pC4 AF
(3.2)
where bi is a constant that is equal to the total non-evaporable water content associated with the
complete hydration of 1 g of the i-th compound. The coefficients can be determined by multilinear regression analysis from a set of wn0 data obtained from presumably completely hydrated
cement samples with different compound compositions. It should be pointed out that accurate
values are difficult to derive due to the fact that truly completely hydrated samples are very
difficult to obtain in practice and that the calculated Bogue weight fractions are only
approximations of the true compound compositions.
Table 3.2 shows the linear regression analysis results from six test series of two major test
programs in the literature as well as the theoretical results of b1 and b2, which were calculated by
assuming the mean formula of dried C-S-H to be C1.7SH1.8, determined by Allen et al. (2007) for
the first time by combining small-angle neutron scattering data and X-ray scattering data without
recourse to drying methods. The experimental coefficients obtained by Powers and Brownyard
(1946) differ quite significantly from those obtained by Copeland et al. (1960). The results from
the former are probably less accurate because of the generally younger and widely variable test
ages. The values of b1, b2, and b3 obtained from Test Series 4, 5 and 6 are very similar, with the
first two also fairly close to the theoretical results. This suggests that the non-evaporable water
52
content of C-S-H obtained at P-dried state is the closest to the chemically combined water
content and that D-drying and oven drying remove more “water of constitution”. The widely
varying values for b4 may be partially due to the incomplete hydration of C4AF in Test Series 4
and 5. The compound (C4AF) had been shown to be the slowest hydrating phase in Portland
cement (Escalante-Garcia 1998). Therefore, the results from Test Series 6 seem most reliable:
wn0 = 0.234 ⋅ pC3S + 0.197 ⋅ pC2 S + 0.509 ⋅ pC3 A + 0.184 ⋅ pC4 AF
(3.3)
It is interesting to note that b1, b2, and b4 are very close while b3 is more than two times higher,
indicating that C3A is the dominating phase for determining the value of wn0. It is shown in
Figure 3.1 that a simple linear regression model with C3A content as the single predictor variable
gives similar goodness of fit (measured by R2 value) as the multi-linear regression model. The
figure also shows that data points for Type III cement are all above the trendlines, suggesting
that higher degrees of hydration were reached for finer particles and that most cements are not
completely hydrated. The R2 values of both models are relatively low, but appear to improve as
samples approach higher degrees of hydration. Unfortunately, the original experimental data of
the more reliable test series (i.e. 4, 5, and 6) were not published. The following simple linear
equation is derived from reproduced data of Test Series 6 using Eq. (3) (cements with the same
compound composition were used to produce only one data point).
wn0 = 0.378 ⋅ pC3 A + 0.193 ( R 2 = 0.85)
(3.4)
The equation suggests that wn0 should range from 0.193 to 0.250 for all cements since the C3A
content is limited to within 15% according to ASTM C150. It is still doubtful whether complete
hydration is possible in normal cement paste or mortar samples, even with a w/c ratio as high as
0.8. Therefore, the validity of the above empirical equations depends on the ultimate degrees of
hydration of the samples.
53
Table 3.2: Coefficients for total non-evaporable water content (P-dried samples)
Test
No. of
Curing
Age
w/c
b1
b2
b3
b4
Source
series cements Condition (years)
0.35
1
100
Saturated
≥0.44 0.187 0.158 0.665 0.213 Powers 1946
-1.31
2
27
Sealed
1
0.4 0.228 0.168 0.429 0.132 Copeland 1960
3
27
Sealed
6.5
0.4 0.234 0.178 0.504 0.158 Copeland 1960
4
27
Sealed
13
0.4 0.230 0.196 0.522 0.109 Copeland 1960
5
24
Sealed
6.5
0.6 0.238 0.198 0.477 0.142 Copeland 1960
6
23
Sealed
6.5
0.8 0.234 0.197 0.509 0.184 Copeland 1960
a
Theoretical Calculations
0.245 0.220
a
: Calculated in this study based on C-S-H formula measured by Allen et al. (2007)
Test Series 3 (w/c=0.4, Age=6.5 years)
Test Series 2 (w/c=0.4, Age= 1 year)
0.25
0.25
Non-evaporable water content (g/g cement)
Experimental data (
0.24
Type III cement)
Multilinear model (R-Square=0.68)
0.24
Linear model (R-Square=0.63)
0.23
0.23
0.22
0.22
0.21
0.21
0.2
0.2
0.19
0.19
0.18
0.18
0.17
0.17
14
Experimental data ( Type III cement)
Multilinear model (R-Square=0.77)
Linear model (R-Square=0.74)
4
6
8
10
12
Tricalcium aluminate content (%)
4
6
8
10
12
Tricalcium aluminate content (%)
14
Figure 3.1: Non-evaporable water content vs. C3A content of cement
(Test data from Verbeck and Foster (1950))
Probably a more reliable way to determine the value of wn0 is by plotting wn(t) as a function
of α(t) (the latter measured by QXRD) and obtaining the best fit slope. For such purpose, Eq.
(1.2) may be rewritten as,
54
wn0 =
wn (t )
α (t )
(3.5)
Parrott et al. (1990) and Escalante-Garcia (2003) studied the correlation between non-evaporable
water content wn(t) and the degree of hydration α(t) measured by QXRD. The best-fit values of
wn0 and the cement chemical compositions are listed in Table 3.3, where it is observed again that
wn0 increases with C3A content. However, the different drying methods used in these two studies
to determine non-evaporable water content (see Table 1.2) and the lack of a conversion factor
make comparisons with previously derived empirical models difficult. Another more reliable
way to determine the value of wn0 is by using ball-milled slurries with very high water to cement
ratios. Mills (1966) adopted two w/c ratios (2.1 and 4.7) and ground the samples until “no
detectable change” in non-evaporable water content was observed, which took up to two months.
The samples were then diluted to a w/c ratio of 12 and stored for about 1 year. Average increases
of 5% and 1% in non-evaporable water content were observed for slurries made with w/c ratios
of 2.1 and 4.7 respectively. Bentz (1995) used a w/c ratio of 3.0 and claimed that “little change in
the non-evaporable water content was observed after the first seven days of grinding” and
assumed complete hydration had been achieved in samples ground for a period exceeding 28
days. The obtained values of wn0 from these two studies together with cement compositions are
also presented in Table 3.3. In this case, we are able to derive the values of wn0 for P-dried
samples according to the previously discussed conversion factor found by Powers and
Brownyard (1946). As shown in the table, it is obvious that Eq. (3.3) significantly
underestimated wn0, especially according to Mills’ test results. As a matter of fact, Mills also
reported that the highest degree of hydration achieved by mortar samples at the age of 1.2 years
was only 83% for a w/c ratio of 0.85, which decreases further as w/c ratio decreases.
55
Table 3.3: Experimental and predicted value of wn0 for different cements
Compound composition (%)
Source
Parrott 1990
(BCA, LOI)
EscalanteGarcia 2003
Mills 1966
Bentz 1995
This Study
C3S
C2S
C3A
C4AF
72
59
39
52.6
50
49.7
48.4
54.7
54.4
70.3
13
13
31
19.4
22.2
25.6
26.0
21.4
19.5
8.5
4
8
10
6.9
10.2
10.6
10.9
8.1
13.7
0
1
11
8
9.4
8.8
7.6
7.9
9.3
5.4
12.8
Experimental results
Predicted results
a
Adjusted
Eq.(3.3) Eq.(3.6)
wn0
R2
wn0
0.252b 0.97b
0.216
0.238
b
b
0.273
0.98
0.225
0.247
0.329b 0.97b
0.218
0.240
0.24
0.96
0.214
0.235
0.25
0.96
0.229
0.252
0.253
0.284
0.235
0.258
0.253
0.284
0.234
0.258
0.226
0.254
0.228
0.251
0.235
0.264
0.245
0.270
0.205
0.2254
a
: Adjusted for the effect of drying method and assuming P-drying as the standard.
: Obtained from reproduced test data.
b
For the lack of better experimental data, it is assumed here that the samples in Test Series 6
(Table 3.2) were about 90% hydrated, and the following equation can be derived by applying a
factor of 1.1 to Eq. (3.3)
wn0 = 0.257 ⋅ pC3S + 0.217 ⋅ pC2 S + 0.560 ⋅ pC3 A + 0.202 ⋅ pC4 AF
(3.6)
After such adjustment, the coefficients for C3S and C2S still agree well with theoretical
calculations (errors are within 5%). In addition, as shown in Table 3.3, values of wn0 predicted
with Eq. (3.6) agree reasonably well with experimental results of ball-milled slurries (errors are
within 10%). The equation shall be further calibrated in the future when more experimental data
become available.
3.4 Specific Volumes of Different Types of Water in Cement Paste
To evaluate chemical shrinkage, it is convenient to assign a hypothetical specific volume
(vn) to the non-evaporable water in a hydrated cement paste (Powers 1946), which can be
56
calculated by attributing the volume changes of all hydration reactions to the volume reduction
of the reacted water only. Since non-evaporable water is defined as the water content of a dried
cement paste, the chemical formulae and densities of hydration products at dried state shall be
used in such calculations. It should be pointed out that vn has no literal significance with its value
depending both on the reactants and the resultants of a chemical reaction and its introduction is
only to facilitate calculation. An example of vn calculated for hydration products of C3S is as
follows
Volume
C3S + 3.1H → C1.7SH1.8 + 1.3CH
change
Molar Mass (g/mol) 228.3
95.5
187.9
96.3
3
3.15
0.998
2.604
2.242 (cm3/g water
Density (g/cm )
Mass (g)
4.0883
reacted)
1
3.3636
1.7247
3
Volume (cm )
1.2979
1.0020
1.2917
0.7693
-0.239
νn
(cm3/g)
0.763
Calculations for C2S hydration yielded vn = 0.781 cm3/g. Accurate values of vn’s for hydration of
other phases in cement cannot be calculated due to uncertainties about their hydration products in
dried states. The specific volume of non-evaporable water of a particular cement paste is
measurable experimentally and should depend on the original cement composition.
The specific volume of a saturated cement paste (or mortar) can be expressed as
v p = ms vs + mc vc + mt vt + v0
(3.7)
mt vt = mn vn + mg vg + mw vw
(3.8)
where
where ms, mc, mt, mn, mg, and mw are the mass fractions (in g per g of saturated sample) of sand
or other inert solid filler (if present), original cement, total water, non-evaporable water, gel
water, and capillary water, respectively, while vs, vc, vt, vn, vg, and vw are their respective specific
volumes (in cm3/g). v0 is the total void (cm3) per gram of sample not filled by water. Most of the
57
parameters in Eqs. (3.7) and (3.8) can be directly determined from experiments, except for the
specific volumes of different types of water (vn, vg, and vw). The specific volume of capillary
water shall be slightly smaller than (and is sometimes assumed to be approximately equal to)
pure free water due to a small amount of dissolved salt. Powers and Brownyard (1946) studied
nearly 200 cement paste and mortar samples produced with cements of many different
compositions and proposed vn = 0.82 cm3/g and vg = 0.90 cm3/g. A later study by Copeland
(1956) using only cement paste samples suggested that vn = 0.74 cm3/g and vg = vw = 0.99 cm3/g.
It should be pointed out that P-drying was used by the former while D-drying was used by the
latter to determine the non-evaporable water content. To minimize the effect of air void content
(v0), which is not taken into account in calculations, in the former study samples were ground to
small granules and re-saturated while in the latter study the paste was prepared in vacuum. By
revisiting the former study, Brouwers (2004, 2005) recently suggested that vg = 0.90 cm3/g is
more likely.
Since debate is still ongoing, it is important to first address the question whether the
specific volume of gel water is lower than that of capillary water. An approach similar to the one
used by Copeland (1956) is adopted here to reanalyze the test data of Powers and Brownyard
(1946) (Tables 5-1 to 5-6 in the publication). As discussed earlier, the separation of nonevaporable water from the hydration products when analyzing the specific volume of a cement
paste, as suggested by Eqs. (3.7) and (3.8) is only to facilitate later calculations and has no
physical significance. A more rigorous way to represent the specific volume of a saturated
cement paste (or mortar) is as follows (ignoring air voids):
v p = ms vs + muc vc + m p′v p′
where
(3.9)
58
 w 
m
muc = (1 − α )mc = 1 − n0  mc = mc − n0
wn
 wn 
(3.10)
m p′ = 1 − ms − muc
(3.11)
muc and mp’ are the mass fractions of the anhydrous cement and the completely hydrated part of
the cement paste, respectively; vp’ is the average specific volume of the completely hydrated
cement paste, which is derivable from Eqs. (3.9) - (3.11), where it has been assumed that the
specific volume of the anhydrous cement is the same as that of the original cement. The values of
wn0 for different cements can be calculated according to Eq. (3.6).
The completely hydrated cement paste can be further divided into two parts: hydration
products (including reacted cement and non-evaporable water) and evaporable water (including
gel water and capillary water). Let f be the mass fraction of evaporable water in the hydrated
paste (me/mp’), then
v p′ = (1 − f )vhp + fve
(3.12)
v p′ = vhp + (ve − vhp ) f
(3.13)
Or
If a least square linear relationship between vp’ and f is obtained, then its intercepts with the lines
f = 0 and f = 1 give the average specific volumes of the hydration products and evaporable water,
respectively. Since the mass fraction of gel water is approximately equal to that of nonevaporable water (Powers 1946), the gel water content in the evaporable water roughly equals
the mass ratio (r) of non-evaporable water to evaporable water. If the specific volume of gel
water was lower than that of capillary water, then the average specific volume of evaporable
water should decrease with increasing r. Test data were separated into three sets according to the
59
values of r (Figure 3.2). The average specific volumes of evaporable water calculated with the
linear fitted parameters obtained from Data Sets I, II, and III were 0.953, 1.001, and 0.961 cm3/g,
respectively, which clearly did not decrease with increasing r. Hence, the assumption that gel
water is denser than capillary water is unjustified. The coefficients of determination (R2 value) of
these fits were relatively low (0.86-0.95) due to the small ranges of data. A linear fit to all test
data resulted in a much higher R2 value, which is shown in Figure 3.2. The average specific
volumes of the hydration products and evaporable water are found to be 0.410 and 0.961 cm3/g,
respectively, in reasonable agreement with those determined by Copeland (1956) (0.398 and
0.990 cm3/g). The results from the later study are more reliable because of the better technique to
eliminate air bubbles (vacuum mixing) and the more complete range of test data (samples at very
Specific volume of hydrated cement pastes
young ages (f > 0.5) were also used).
0.7
0.65
y = 0.5513x + 0.4099
R2 = 0.9781
0.6
0.55
0.5
Data Set I, r < 0.5
Data Set II, 0.5 < r < 0.75
Data Set III, r > 0.75
0.45
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Weight fraction of evaporable water in hydrated cement pastes (f)
Figure 3.2: Variation of specific volume of hydrated cement paste with evaporable water content
(r ≈ gel water content in evaporable water, test data from Powers and Brownyard (1946))
60
When the specific volume of gel water is assumed to be equal to that of capillary water, Eq.
(3.8) can be rewritten as
mt vt = mn vn + ( mt − mn )vw
(3.14)
or
vw − vn =
vw − vt
mn / mt
(3.15)
or
vt = vw − (vw − vn )
mn
mt
(3.16)
The previously proposed vn = 0.72 cm3/g finds its origin in the reported average value of (vw - vn)
of 0.279 cm3/g calculated according to Eq. (3.15) by Powers and Brownyard (1946) assuming vw
= 1 cm3/g. On the other hand, vn = 0.74 cm3/g finds its origin in a least square fit equation
obtained by Copeland (1956) according to Eq. (3.16)
vt = 0.988 − (0.988 − 0.737)
mn
mt
(3.17)
Note that D-drying was used to determine mn in Eq. (3.17). Assume the non-evaporable water
content measured with D-dried samples is 8% smaller than that measured with P-dried samples
(Brouwers 2004), then for P-dried condition, the equation becomes
vt = 0.988 − (0.988 − 0.757)
mn
mt
(3.18)
Hence vn = 0.757 cm3/g, which is very close to the theoretically calculated value for the
hydration of C3S (0.763 cm3/g). If the same analytical method (least square fit) were applied to
the test data of Powers and Brownyard (1946), the following relationship is derived
61
vt = 0.986 − (0.986 − 0.747)
mn
mt
(3.19)
Hence vn = 0.747 cm3/g, which is in good agreement with the value obtained from Eq. (3.18). An
average value of 0.752 cm3/g is adopted in this study. It is also noted that the fitted values of
specific volume of capillary water (which is the same as that of gel water and evaporable water)
in Eqs. (3.18) and (3.19) are in almost perfect agreement with that find by Copeland (1956)
according to Eq. (3.13). An average result of 0.988 cm3/g is adopted here.
3.5 Correlation between Chemical Shrinkage and Non-evaporable Water
In summary, both gel water and capillary water were found to have the same specific
volume of about 0.988 cm3/g, slightly lower than pure free water, probably due to small amounts
of dissolved salt. The average specific volume of non-evaporable water was found to be
approximately 0.752 cm3/g, much lower than that of capillary water. Therefore, the chemical
shrinkage accompanying cement hydration is mainly caused by the transformation of capillary
water to non-evaporable water. Fig 3.3 illustrates the volume change process during the
hydration of one unit volume of cement. As discussed in Section 3.3, the total weight of a cement
paste may change during the hydration process as a result of bleeding and water imbibitions.
However, in a typical chemical shrinkage test (including those in this study), both bled water and
imbibed water are part of the entire volume being monitored and the total mass of the system is
truly constant. Therefore, chemical shrinkage can be related to the total mass of non-evaporable
water by the following equation,
CS (t ) = wn (t ) ⋅ (vw − vn )
(3.20)
62
By substituting the specific volumes of capillary water and non
non-evaporable
evaporable water,
water the
normalization factor (CS0) between chemical shrinkage and degree of hydration can be related to
the normalization factor (wn0) between non
non-evaporable
evaporable water and degree of hydration by the
following equation,
CS 0 = wn0 (vw − vn ) = 0.236wn0
(3.21)
The following empirical model may be obtained by substituting Eq. (3.6) into Eq. (3.21):
CS 0 = 0.0607 ⋅ pC3 S + 0.0511 ⋅ pC2 S + 0.1321 ⋅ pC3 A + 0.0478 ⋅ pC4 AF
(3.22)
Figure 3.3:: Total volume change during cement hydration
(Vc and Vw are the total volumes of cement and water, respectively, before hydration; Vuc, Vhc,
and Vhp are the volumes of anhydrous cement, reacted cement, and hydration product,
respectively, after hydration; Vn, Vg, and Vw’ are the volumes of non-evaporable
evaporable water, gel water,
water
and capillary water, respectively, after hydration
hydration;; other parameters are the same as defined
previously)
63
3.6 Model Application
The primary application of the empirical model developed in this chapter is to convert
chemical shrinkage test data to the degree of hydration of cement such that cement hydration
kinetics can be studied. It should be noted that Eqs. (3.20) and (3.21) are derived for ambient
curing conditions. The specific volume (or density) of water is known to vary with both
temperature and pressure. As suggested by Eq. (3.21), a slight change in vw or vn is magnified
more than 4 times in terms of its effect on the difference between the two. Therefore, for samples
cured at non-ambient conditions, the effect of pressure and temperature on the normalization
factor (CS0) should not be ignored. In the case of capillary water (similar to free water), neither
the bulk modulus nor the volumetric expansion coefficient is constant. But those of fresh water
have been successfully modeled (Bahadori 2009) with very good accuracy for temperatures
ranging from 0 to 50 °C and pressures ranging from 0.1 to 55 MPa. In the case of nonevaporable water, since vn is a hypothetical term, it is difficult to model its dependence on
pressure and temperature. It is assumed here that the bulk modulus of non-evaporable water is
the same as that of free water compressed to the same specific volume and that its dependence on
pressure may be ignored since non-evaporable water is part of the solid state. A value of 10.6
GPa is derived by extrapolating the data obtained from NIST Chemistry WebBook (2011), where
the lowest specific volume given was 0.813 cm3/g.
By assuming the bulk modulus of capillary water to be the same as that of fresh water
modeled by Bahadori and Vuthaluru (2009) and a constant bulk modulus of 10.6 GPa for nonevaporable water, the variation of specific volume (∆v) with that of pressure (∆P) can be
calculated with the following equation
64
 ∆P 
∆v = − v 

 K 
(3.23)
where v is the reference specific volume and K is the bulk modulus. The specific volumes of
capillary water (vw) and non-evaporable water (vn) at the atmospheric pressure can be used to
estimate their values at other curing pressures using Eq. (3.23). Note that those of the capillary
water have to be estimated numerically since K is not constant. Due to difficulties associated
with estimating the dependence of vn on temperature, a more approximate method is used in this
study to calculate CS0 for different curing temperatures. Zhang et al. (2010) found that CS0 at
40 °C and 60 °C were about 87.4% and 72.5% of that at 25 °C, respectively, suggesting an
almost linear reduction. In this study, it was assumed that CS0 decreases linearly at a rate of
0.783% per °C from the value obtained at the ambient temperature. The calculated values of CS0
for different cement at different curing conditions are listed in Table 3.4. The coefficient for C2F
was obtained by assuming it causes the same amount chemical shrinkage as C4AF on the same
mass basis. These results are used in Chapters 4, 5, and 6 to derive cement hydration kinetics
data from chemical shrinkage tests.
Table 3.4: Estimated values of CS0 for different cement under different curing conditions
Temp.
Press.
νw
vn
CS0
(mL/100g
cement)
°F
°C
psi
MPa
cm3/g
cm3/g
A
C
G
H-P
H-I
H-II
atm.
atm.
100
0.69
1000
6.9
77
25
1900
13.1
0.988
0.752
5.914
5.505
5.771
5.140
5.315
5.320
0.9877
0.7520
5.906
5.498
5.763
5.133
5.308
5.313
0.9849
0.7515
5.256
5.261
0.9823
0.7511
5.207
5.212
2500
17.2
5000
34.5
0.9805
0.7508
5.756
5.358
5.617
5.003
5.173
5.178
0.9733
0.7496
5.606
5.218
5.470
4.872
5.038
5.042
7500
51.7
105
40.6
100
0.69
140
60
100
0.69
0.9665
0.7483
5.468
5.090
5.335
4.752
4.914
4.918
5.159
4.922
5.059
4.529
4.618
4.622
4.262
4.086
4.183
3.749
3.812
3.815
65
3.7 Summary
The correlations between two important indirect methods used to measure the degree of
hydration and hydration kinetics of Portland cement, namely the non-evaporable water and the
chemical shrinkage tests were reviewed in this study. Critical parameters that can be used to
convert test data to degree of hydration are identified. The reliability of existing empirical
models used to estimate these parameters is investigated and new models are proposed. The main
findings from this study include the following.
1. The non-evaporable water content of hydrated cement depends on the specific test method
used; the values obtained by the loss on ignition method performed on P-dried samples are
probably the closest to the chemically combined water content of the sample.
2. The specific volume of gel water in a saturated cement paste is roughly the same as that of
capillary water, which is approximately 0.988 cm3/g.
3. The average specific volume of non-evaporable water in hydrated cement is found to be
0.757 and 0.747 cm3/g, respectively, according to test results of Powers and Brownyard
(1946) and those of Copeland (1956).
4. The following equations may be used to estimate the total non-evaporable water content for
completely hydration cement:
wn0 = 0.257 ⋅ pC3 S + 0.217 ⋅ pC2 S + 0.560 ⋅ pC3 A + 0.202 ⋅ pC4 AF
5. The following equations may be used to estimate the total the total chemical shrinkage for
completely hydrated cement at ambient temperatures (approximately 25 °C):
CS 0 = 0.0607 ⋅ pC3S + 0.0511 ⋅ pC2 S + 0.1321 ⋅ pC3 A + 0.0478 ⋅ pC4 AF
66
CHAPTER 4 : MODELING THE EFFECT OF CURING
TEMPERATURE AND PRESSURE ON CEMENT HYDRATION
KINETICS
4.1 Introduction
Cement hydration kinetics is typically represented by two types of curves (Figure 4.1):
degree of hydration vs. time (defined here as the integral curve) and rate of hydration vs. time
(defined here as the derivative curve). The hydration process may be classified into five periods
according to the derivative curve: (1) initial reaction, (2) induction (dormant) period, (3)
acceleration period, (4) deceleration period, and (5) steady state. In the past few decades,
significant efforts have been devoted to model the hydration kinetics of cement. Although
progress has been made, recent reviews showed that all of the models developed to date have
0.025
0.5
0.02
0.4
0.01
4
3
0.015
0.3
1
0.2
2
0.005
Degree of hydration
Rate of hydration (/h)
their limitations and a universally accepted model is still absent (Xie 2011, Thomas 2011).
0.1
5
0
0
10
20
30
40
Time (h)
50
60
0
70
Figure 4.1: Representative hydration kinetic curves of Class H-II cement (w/c = 0.38)
67
The main influencing factors of cement hydration kinetics include internal factors such as
properties of cement (chemical composition and particle size distribution) and w/c ratio, and
external factors such as curing conditions. This chapter focuses on modeling the external factors.
More specifically, the goal is to develop a universal model that can be readily applied to any
other hydration kinetics model to account for the effect of curing temperature and pressure on
hydration. This is achieved by representing hydration kinetics with functions whose exact
expressions are not known. For verification of the model, experimentally obtained hydration
kinetics curves at a reference curing condition were used to predict hydration kinetics curves at
other curing conditions by simple coordinate transformations. The advantages of the proposed
approach include: (1) the model developed here can be readily combined with other models
developed to account for the internal factors. (2) The applicability and reliability of the model
can be checked directly for all stages of hydration by comparing the predicted curves (obtained
by transforming the experimental curves of the reference curing condition) with the experimental
curves at the curing conditions to be simulated. (3) The model can also be conveniently used to
predict the effect of curing temperature and pressure on other properties of cement that have a
one-to-one relation with the degree of hydration.
The hydration kinetics of cement during early periods is traditionally measured by
isothermal calorimetry. Standard calorimeters allow different curing temperatures to be applied,
but not different pressures. Consequently, the effect of curing temperature on cement hydration
kinetics is relatively well understood today while the effect of curing pressure is still unclear.
The new chemical shrinkage test method developed in this study allows cement hydration
kinetics to be measured at both different temperatures and different pressures. These test data are
used to verify the new approach of modeling cement hydration kinetics proposed in this chapter.
68
4.2 Preliminary Analysis of Test Data
4.2.1 Chemical shrinkage data
Figure 4.2 shows some representative results of chemical shrinkage tests performed at
different curing temperatures as well as different curing pressures (Series I of pressure cell tests).
Total chemical shrinkage is found to increase considerably with increasing curing temperature at
early age, which is consistent with the current state of knowledge (Mounanga 2006). The effect
of curing pressure on total chemical shrinkage, however, is too small to be evaluated accurately
due to the natural errors in test results. Figure 4.3 shows the test results for a much higher
pressure range (Series II of pressure cell tests). Total chemical shrinkage is also found to increase
with increasing curing pressure at early age. Note that the test results obtained at ambient
temperatures also reflect temperature fluctuations due to inadequate temperature control scheme
adopted in this study. The fact that similar chemical shrinkage results were obtained at 17.2 and
34.5 MPa is probably due to the fact that the latter was obtained at a slightly lower lab
temperature.
Chemical Shrinkage (mL/g cement)
0.03
0.025
60 °C
0.02
0.015
40.6 °C
24 °C
0.01
0.69MPa
6.9MPa
13.1MPa
0.005
0
0
10
20
30
40
50
Time (h)
Figure 4.2: Effect of curing temperature and pressure on total chemical shrinkage
(Class H-II cement, w/c = 0.4)
69
Chemical shrinkage (mL/g cement)
0.03
0.025
0.02
0.69MPa
17.2MPa
34.5MPa
51.7MPa
0.015
0.01
0.005
0
0
10
20
30
40
Time (h)
50
60
70
Figure 4.3: Effect of curing pressure on total chemical shrinkage
(Class H-II cement, w/c = 0.38, ambient temperatures)
4.2.2 Hydration kinetics data
As shown in Table 3.4, the correlation factors between chemical shrinkage and the degree
of hydration of cement vary with curing conditions. Therefore, the effect of curing temperature
and pressure on cement hydration kinetics is slightly different from their effect on chemical
shrinkage. Figure 4.4 shows the effect of curing temperature on hydration kinetics (both the
integral curves and the derivative curves) for the same curing pressure of 13.1 MPa. Test data
were smoothed using the method discussed in Section 2.4.1. It is observed that the total degree of
hydration attained at any given time during the first two days increases with increasing curing
temperature. At higher curing temperatures, the cement hydration rate is found to be greatly
accelerated during the pre-peak period, resulting in a shorter acceleratory period. After the peak,
the hydration rate also decreases faster at higher curing temperatures, resulting in a slower
hydration rate during the later stage of the deceleration period. A similar phenomenon has been
70
observed in isothermal calorimetry studies of cement hydration kinetics (Mounanga 2006, Poole
2007, Reinhardt 1982, De Schutter 1995, Ma 1994). Figure 4.5 shows the effect of curing
pressure on hydration kinetics. It is found that curing pressure has a similar effect on hydration
kinetics as curing temperature. However, it appears that a relatively large increase in curing
pressure is only equivalent to a small increase in curing temperature in terms of their effect on
hydration kinetics. This is consistent with another study (Scherer 2010) that investigated the
viscosity evolution of oilwell cement at very early stages.
0.7
40.6°C
Rate of hydration (/h)
Degree of hydration
24°C
0.15
0.6
0.5
0.4
0.3
0.2
24°C
60°C
0.1
0.05
40.6°C
0.1
60°C
0
0
10
20
Time (h)
30
40
0
0
10
20
Time (h)
Figure 4.4: Effect of curing temperature on cement hydration kinetics
(Class H-II cement, w/c = 0.4, curing pressure = 13.1 MPa)
30
40
71
0.5
0.05
0.04
Rate of hydration (/h)
Degree of hydration
0.4
0.3
0.2
0.69MPa
17.2MPa
34.5MPa
51.7MPa
0.1
0
0.69MPa
17.2MPa
34.5MPa
51.7MPa
0
10
20
Time (h)
30
0.03
0.02
0.01
40
0
0
10
20
Time (h)
30
40
Figure 4.5: Effect of curing pressure on cement hydration kinetics
(Class H-II cement, w/c = 0.38, ambient temperatures)
Due to the particular characteristics of cement hydration, the hydration rate is strongly
dependent on the total amount of hydration products generated on the surface of cement particles
as well as in the inter-particle spaces. Since the total amount of hydration products is
proportional to the degree of hydration, it is more appropriate to present the rate of hydration as a
function of the degree of hydration when investigating the hydration mechanisms of cement.
Figure 4.6 and Figure 4.7 show the plots of rate of hydration vs. degree of hydration (defined
here as the differential equation curves) before and after normalization. The differential equation
curves obtained at different curing temperatures and pressures appear to converge to a universal
curve when normalized. Other studies have reported similar results when hydration kinetics was
measured by isothermal calorimetry at different curing temperatures (Reinhardt 1982, De
Schutter 1995). Therefore, it appears that, for the range studied here, both curing temperature and
curing pressure have only a kinetic effect on cement hydration and do not change reaction
mechanisms. The effect of curing temperature and pressure on cement hydration kinetics is
represented by a more or less constant scale factor on hydration rate as a function of degree of
72
hydration and the scale factor is approximately equal to the ratio of the peak hydration rates for
different curing conditions.
24°C
0.14
40.6°C
0.12
60°C
24°C
Normalized rate of hydration
Rate of hydration (/h)
1
0.16
0.1
0.08
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
Degree of hydration
60°C
0.6
0.4
0.2
0
0.5
40.6°C
0.8
0
0.1
0.2
0.3
0.4
Degree of hydration
0.5
Figure 4.6: Effect of curing temperature on hydration rate as a function of degree of hydration
(Class H-II cement, w/c = 0.4, curing pressure = 13.1 MPa)
0.05
1
Rate of hydration (/h)
0.04
Normalized rate of hydration
0.69MPa
17.2MPa
34.5MPa
51.7MPa
0.03
0.02
0.01
0
0
0.1
0.2
0.3
0.4
Degree of hydration
0.5
0.8
0.6
0.4
0.69MPa
17.2MPa
34.5MPa
51.7MPa
0.2
0
0
0.1
0.2
0.3
0.4
Degree of hydration
0.5
Figure 4.7: Effect of curing pressure on hydration rate as a function of degree of hydration
(Class H-II cement, w/c = 0.38, ambient temperatures)
73
The Class H cement used in this study is of high sulfate-resistance (HSR) grade that has
virtually no C3A content. The derivative curves (and differential equation curves) of cement that
contains C3A typically have two peaks, with the first one attributed to the hydration of C3S and
the second one attributed to the hydration of C3A. Figure 4.8 shows the differential equation
curves of Class C cement before and after normalization, which further confirms the previous
findings. As a matter of fact, similar convergences of differential equation curves obtained at
different curing pressures are observed for all the cement used in this study. These results
suggest that the pressure sensitivities of different phases in cement are very similar and that for
the same cement slurry cured at different curing pressures roughly the same degrees of hydration
were attained at the end of the induction period as well as at the peak hydration rates. The
normalized differential equation curves can be averaged to further reduce test data oscillations.
Figure 4.9 shows the average test results of different cement. The curves of Class A and G
cement are very similar due to their similar composition and particle size distributions.
Interestingly, those of Class H-P and H-II cement are also very similar despite their drastically
different C3S and C2S contents. The degree of hydration that corresponds to the first hydration
peak is very similar (approximately 0.1) for Class A, G, H-P, and H-II cement. The hydration
rate reaches its peaks at much higher degrees of hydration for Class C cement probably because
of its much higher surface area (finer particle size). The particle size distribution of Class H-I
cement was not measured.
Unfortunately, it is difficult to obtain reliable derivative curves and differential equation
curves of other types of cement at high curing temperatures using the polynomial fit method
discussed in Section 2.4.1 partially because no repeated tests were performed to average the test
74
results. The effect of curing temperature on cement hydration kinetics is further investigated in
the next chapter using the isothermal calorimetry test data.
0.69MPa
17.2MPa
34.5MPa
51.7MPa
0.1
Normalized rate of hydration
1
Rate of hydration (/h)
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
Degree of hydration
0.8
0.6
0.4
0.2
0
0.8
0
0.2
0.4
0.6
Degree of hydration
0.8
Figure 4.8: Effect of curing pressure on hydration rate as a function of degree of hydration
(Class C cement, w/c = 0.56, ambient temperatures)
1
A
C
G
0.8
Normalized rate of hydration
Normalized rate of hydration
1
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Degree of hydration
0.8
H-P
H-I
H-II
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
Degree of hydration
0.5
Figure 4.9: Normalized hydration rate as a function of degree of hydration for different cement
(w/c ratios of Class A, C, G, H cement are 0.46, 0.56, 0.44, and 0.38, respectively)
75
4.3 Model Development
4.3.1 Model formulation
Cement hydration is essentially an aggregation of a number of chemical reactions.
Therefore, the dependence of hydration rate on temperature and pressure can be modeled by
chemical kinetics theory. The dependencies of a reaction rate constant on temperature and
pressure are associated with the activation energy and the activation volume, respectively
(Laidler 1987, IUPAC 2010):
 ∂ ln k 
Ea

 =
2
 ∂ (T )  P RT
(4.1)
∆V
 ∂ ln k 

 =−
RT
 ∂P T
‡
(4.2)
where k is the specific reaction rate; T is the absolute temperature (K); P is the pressure (Pa); Ea
is the activation energy (J/mol); R is the gas constant (8.314 J/(mol·K)); and ∆V‡ is the activation
volume (m3/mol). Note that Eq. (4.1) is equivalent to,
 ∂ ln k 
Ea

 =−
R
 ∂ (1 / T )  P
(4.3)
If we assume the activation energy is independent of temperature, then for reactions occurring at
a constant pressure (Pr), the reaction rate (k’) at any temperature T can be related to that (kr) at a
reference temperature Tr by,
E
k ' = kr exp  a
 R

 1 1 
 −   = kr C (T )
 Tr T  
(4.4)
where C(T) is the scale factor on reaction rate due to temperature change. Similarly, for reactions
occurring at a constant temperature (T), the reaction rate (k) at any pressure P can be related to
that (k’) at the reference pressure Pr by,
76
 ∆V ‡

k = k 'exp 
( Pr − P )  = k ' C ( P )
 RT

(4.5)
where C(P) is the scale factor on reaction rate due to pressure change. Combining Eq. (4.4) and
Eq. (4.5), the reaction rate at any condition can be derived from that of a reference condition by a
scale factor C(T, P) as follows,
k = k r C (T ) C ( P ) = k r C (T , P )
(4.6)
 E  1 1  ∆V ‡  Pr P  
C (T , P ) = exp  a  −  +
 −  
 R T T
R
 T T 
r



(4.7)
where
As discussed earlier, cement hydration kinetics may be represented by three different types
of curves, namely degree of hydration as a function of time, rate of hydration as a function of
time, and rate of hydration as a function of degree of hydration. Since the exact expressions of
these curves are not known, the following equations may be used to represent cement hydration
kinetics at a reference curing condition:
α r = f ( tr )
(4.8)
dα r
= g ( tr )
dtr
(4.9)
dα r
= z (α r )
dtr
(4.10)
where tr and αr are the time and the degree of hydration of cement at the reference curing
condition (Tr, Pr), respectively. Eqs. (4.8) - (4.10) are essentially the different forms of one
function. It is obvious that g(x) is the derivative of f(x) and Eq. (4.8) is simply the solution to the
differential equation (4.10). Three different functions have to be used here because none of them
is known explicitly. According to the test results presented in the previous section and the
77
chemical kinetics theory, the differential equation curve for any curing condition (T, P) should
differ from the reference curve by only a scale factor C on the y-axis, i.e.
dα
= C ( T , P ) ⋅ z (α )
dt
(4.11)
dα
= z (α )
d ( C (T , P ) ⋅ t )
(4.12)
or
Apparently, the solution to the above differential equation is
α = f ( C (T , P ) ⋅ t )
(4.13)
which happens to be the expression of the integral curve for the new curing condition (T, P).
Therefore, the derivative of Eq. (4.11) gives the expression for the rate of hydration:
dα
= C (T , P ) ⋅ g ( C (T , P ) ⋅ t )
dt
(4.14)
Eqs. (4.11), (4.13), and (4.14) can be used to derive the hydration kinetics of any curing
condition from the reference condition when the kinetics is represented by the differential
equation curve (Eq. (4.10)), the integral curve (Eq. (4.8)), and the derivative curve (Eq. (4.9)),
respectively.
4.3.2 Significance of the scale factor C(T, P)
Since the scale factor C(T, P) is essentially the proportionality constant (y-axis) between
the differential equation curve of a non-reference condition and that of the reference condition, it
can be calculated using any points on the two differential equation curves as long as they have
the same x value (i.e. degree of hydration). The differential equation curve is usually not directly
obtained from experiments, but it is easily derivable from both the integral curve and the
78
derivative curve, whichever is available. A more direct way to calculate the scale factor is by
using the characteristic values of the derivative curve, such as the peak hydration rate at the end
of the acceleration period. The reason why this approach works is because when a derivative
curve is converted to a differential equation curve, only the x values of the curve change, the y
values (including those of the characteristic points) remain the same.
According to Eqs. (4.8) and (4.13), if the time it took for a sample cured at the reference
condition to reach a certain degree of hydration is t, then it would take t/C(T, P) for a sample
cured at a different condition to reach the same degree of hydration. Therefore, the equivalent
time for samples cured at different conditions to reach the same degree of hydration can also be
used to derive the scale factor C(T, P) and model the effect of curing condition on hydration. To
obtain the equivalent time experimentally, this certain degree of hydration has to be associated
with some measurable characteristics of the cement. Correspondingly, the derived model
parameters can also be used to predict the effect of curing conditions on these measurable
characteristics of cements. For example, similar concepts have been successfully adopted to
predict the effect of curing temperature and pressure on the limit of pumpability (Scherer 2010),
and the effect of curing temperature on the setting time (Pinto 1999, García 2008, Zhang 2010),
of cements by assuming each of the quantities corresponds with a fixed degree of hydration.
4.4 Effect of Curing Temperature on Hydration Kinetics
As discussed earlier, the temperature dependence of cement hydration rate is described by
its activation energy (Ea), which can be obtained by performing a linear fit using hydration rates
obtained at different curing temperatures for the same degree of hydration according to Eq. (4.3).
79
Equivalently, the linear fit can also be performed using the following model derived from Eq.
(4.4):
Ea =
R ln ( C (T ) )
 1 1
 − 
 Tr T 
(4.15)
where
 dα / dt 
C (T ) = 

 dα r / dtr α =α r
(4.16)
An important source of error in estimating Ea is that the estimated points of the “same degree of
hydration” are not exact because the actual degrees of hydration achieved at the assumed zero
point (one hour after mixing) may not be the same for different curing conditions. In addition,
due to the presence of different clinker phases (which hydrate at different rates and have
different activation energies) in cement, its activation energy does not remain constant as curing
age increases. Several different methods have been used to derive the “apparent” activation
energy of cement (i.e. a representative value) from experimental data (Ma 1994, Kada-Benameur
2000, D’Aloia 2002, Mounanga 2006, Poole 2007). An incremental calculation method, i.e.
performing a least square fit for each step increase in degree of hydration, is usually used to
evaluate the dependence of Ea on degree of hydration (Poole 2007).
Series I of pressure cell test results were used to calculate the apparent activation energy of
Class H-II cement. The influence of curing pressure on hydration kinetics is ignored for this test
series as it was found to be smaller than the random errors caused by temperature fluctuations
and other factors. Figure 4.10 show the activation energy of Class H-II cement calculated
incrementally based on test data obtained at the curing pressure of 13.1 MPa. Since the early
hydration mechanism of cement (the first two periods in Figure 4.1) is still not clearly
80
understood, the hydration rate at very early ages (t < 2h) may not follow Eq. (4.1) and the
activation energy obtained during this period (typically α < 0.03) probably has no physical
significance. As shown in Figure 4.10, activation energy remained relatively constant between
approximately α = 0.03 and α = 0.4, consistent with previous studies (Kada-Benameur 2000,
Poole 2007). It should be noted that the activation energy obtained at later ages also may not be
accurate because the hydration rate may be too low to be measured accurately. The activation
energies for other types of cement are not calculated incrementally due to difficulties of
obtaining reliable differential equation curves at high temperatures.
Activation energy (kJ/mol)
60
50
40
30
20
Class H-II cement
10
0
0
0.1
0.2
0.3
Degree of hydration
0.4
0.5
Figure 4.10: Variation of activation energy with degree of hydration
As shown in Eq. (4.16), the ratio of hydration rates (i.e. the scale factor) due to temperature
increase can be calculated at any point of the “same degree of hydration”. Similar to the
activation energy, the scale factor is also expected to vary with curing age. To obtain a
representative value, one can simply substitute the peak hydration rates at different curing
temperatures into Eq. (4.16), since they approximately correspond with the same degree of
81
hydration (see Figure 4.6). Alternatively, the scale factor can also be calculated using the
equivalent age concept. As there are probably different mechanisms governing the first two
stages of hydration, it is better to offset the zero time to the onset of the accelerating stage, hence:
C (T ) =
(t
peak
− to )
r
t peak − to
(4.17)
where tpeak is the time at which the hydration rate reaches its peak and to is the offset time (i.e. the
starting point of the accelerating stage); the subscript r represents the reference condition. Both
the peak hydration rates and the characteristic times were estimated by differentiating a sixth
order polynomial fit to test data from the beginning to a point slightly past the peak hydration
rate. The results are listed in Table 4.1. The differences between the obtained peak hydration
rates compared with a linear fit method (to test data 0.5 hours before and after the peak) are
within 2% and 6% for ambient and high-temperature conditions, respectively. The scale factors
calculated with Eq. (4.16) were found to be slightly higher than those calculated with Eq. (4.17),
especially at higher temperatures. It is probably more reliable to derive the scale factors from the
peak hydration rates because the characteristic times are very difficult to determine accurately.
Figure 4.11 shows the linear regression analysis results used to estimate the apparent
activation energies for the different types of cement according to Eq. (4.15). The estimated
values, which were found to increase with increasing C3A content, are shown in Table 4.1. The
results are consistent with other studies that found the activation energy of calcium aluminate
cement to be much higher than that of Portland cement (Bushnell-Watson 1987, Banfill 1995).
Class H-I cement was only tested at ambient temperatures; its activation energy may be assumed
to be the same as that of Class H-II cement due to their similar compositions.
82
Table 4.1: Scale factors for different curing conditions and estimated activation energy
(Characteristic times and peak hydration rates were obtained by a polynomial fit)
Temp. Press.
t
Test No. o
(°C) (MPa)
(h)
24-Ib
0.69
24-I
24a
6.9
24-II/III
13.1
0.69
24-IV
40-I
6.9
40-II/III
40.6
13.1
0.69
60-I
6.9
60-II/III
13.1
60-IV
0.69
24-Ib
24a
6.9
24-II/III
24.4
40.6
60
26.9
40.6
60
25
40.6
60
25.6
40.6
60
13.1
0.69
0.69
0.69
0.69
0.69
0.69
0.69
0.69
0.69
0.69
0.69
0.69
24-IV
A-1b
A-4
A-5
C-1b
C-4
C-5
G-1b
G-4
G-5
H-P-1b
H-P-4
H-P-5
60
a
40-IV
2.75
2.88
2.63
2.61
2.97
2.89
2.70
2.77
2.00
2.21
1.85
1.71
1.81
2.10
1.68
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.82
2.73
2.53
2.53
2.59
1.65
1.46
1.34
1.77
1.32
1.20
1.74
1.35
1.32
2.16
1.56
1.53
C(T) (dα/dt)peak C(T)
tpeak
Ea
Cement
(h-1)
Eq.(4.16)
(h) Eq.(4.17)
(kJ/mol)
10.6
10.33
9.62
9.13
10.53
10.20
9.98
9.52
5.25
4.86
4.91
4.48
4.53
4.54
4.46
3.56
3.67
3.42
3.46
3.61
3.62
3.42
3.41
11.14
10.52
10.10
10.56
10.56
8.76
4.41
2.37
8.07
3.99
2.65
7.87
4.17
2.67
8.38
4.14
3.27
1
1.05
1.12
1.20
1.04
1.07
1.08
1.16
2.42
2.96
2.57
2.83
2.89
3.22
2.82
5.03
4.70
5.53
5.38
4.88
4.85
5.53
5.57
1
1.07
1.10
1.04
1.04
1
2.41
6.90
1
2.36
4.34
1
2.17
4.54
1
2.43
3.57
0.0214
0.0216
0.0228
0.0229
0.0217
0.0219
0.0226
0.0234
0.0580
0.0618
0.0564
0.0617
0.0620
0.0636
0.0598
0.1462
0.1483
0.1567
0.1567
0.1580
0.1532
0.1626
0.1598
0.0232
0.0239
0.0244
0.0240
0.0232
0.0308
0.0975
0.2980
0.0576
0.1314
0.3996
0.0356
0.0972
0.2965
0.0241
0.0637
0.1427
: Estimated average lab temperature
: Reference tests used to calculated the scale factor
b
1
1.01
1.07
1.07
1.01
1.03
1.06
1.10
2.71
2.89
2.64
2.89
2.90
2.98
2.80
6.84
6.94
7.33
7.33
7.39
7.16
7.61
7.47
1
1.03
1.05
1.03
1.00
1
3.16
9.66
1
2.28
6.93
1
2.73
8.33
1
2.64
5.92
H-II
44.3
H-I
A
52.6
C
48.8
G
50.0
H-P
42.5
83
Figure 4.11: Linear regression analyses showing the temperature dependence of the scale factor
C(T) for different cements
4.5 Effect of Curing Pressure on Hydration Kinetics
The activation volume (∆V‡) is used to describe the pressure dependence of hydration rate.
According to Eq. (4.5), ∆V‡ can be calculated as follows for each pressure change from Pr to P
for a constant temperature (T) process:
∆V ‡ =
RT ln ( C ( P ) )
( Pr − P )
(4.18)
The scale factor C(P) can be estimated in the same way as C(T), using Eqs. (4.16) and
(4.17). Since all the tests used for studying the effect of curing pressure (see Series II of pressure
cell tests, Table 2.5) were performed at ambient temperatures, the derivative curves (and the
characteristic times) can be directly derived from experimental data. To reduce the effect of
small data oscillations on test results, the peak hydration rate was obtained using a linear fit to
test data (the integral curve) 0.5 hours before and after the peak. Note that the highest peak was
chosen for curves with double peaks. The relevant test results are shown in Table 4.2. The scale
84
factors calculated with the two different methods were again found to be similar. However, due
to fluctuations of the lab temperature, the values derived directly from experimental data are to
some extent the combined effects of both temperature and pressure changes (i.e. the product of
C(T) and C(P) as shown in Eq. (4.6)). In order to separate these two influencing factors, the
effect of temperature fluctuations on the scale factors can be estimated according to Eq. (4.4)
using the previously derived activation energies. The scale factors calculated from peak
hydration rates were corrected for temperature fluctuations and used to derive the activation
volumes. Figure 4.12 shows the linear regression analysis results used to estimate the activation
volumes for the different types of cement. Note that only a general linear fit line was shown for
similar data sets. It was found that the activation volumes of Class C, H-P, H-I, and H-II cements
were almost the same, but higher than those of Class A and G cements (Table 4.2). The former
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
ln(C(P))
ln(C(P))
values also agree well with that obtained by Scherer et al. (2010)
0.3
0.2
0.1
Class A
Class G
0.3
0.2
Class H-P
0.1
Class H-I
Class H-II
Linear fit
0
-60
-40
-20
Pr - P (MPa)
Class C
Linear fit
0
0
-60
-40
-20
Pr - P (MPa)
0
Figure 4.12: Linear regression analyses showing the pressure dependence of the scale factor C(P)
for different cement
85
Table 4.2: Scale factors for different curing conditions and estimated activation volume
(Characteristic times were obtained directly from experimental derivative curves)
Test No.
A-1
A-2
A-3
A-4
C-1
C-2
C-3
C-4
G-1
G-2
G-3
G-4
H-P-1
H-P-2
H-P-3
H-P-4
H-I-1
H-I-2
H-I-3
H-I-4
H-II-1
H-II-2
H-II-3
H-II-4
Temp.
(°C)
24.4
22.8
25
24.4
26.9
27.5
25
25.6
25
24.7
23.1
25
25.6
22.2
23.9
26.1
25
25
24.7
26.1
26.1
26.7
to
(h)
tpeak C(T,P) (dα/dt)peak C(T,P)
∆V‡
C(T) C(P)
-1
(h )
Eq. (4.16)
(cm3/mol)
(h) Eq.(4.17)
1.42 8.3
1.43 7.9
1.28 5.7
1.34 5.3
1.63 7.6
1.12
6
1.31 5.4
1.08 4.3
1.35 7.9
1.30 6.3
1.35 6.1
1.26 4.6
2.14 8.9
2.09 7.3
1.78 5.8
1.65 4.6
3.04 12
2.59
9
2.59
9
2.28 6.8
2.92 10.1
2.81 8.5
2.76 8.3
2.10 5.5
1
1.06
1.56
1.74
1
1.22
1.46
1.85
1
1.31
1.38
1.96
1
1.22
1.53
1.93
1
1.33
1.33
1.76
1
1.19
1.22
1.84
0.03079
0.03156
0.04553
0.04794
0.05875
0.07182
0.07845
0.09639
0.03618
0.04512
0.04294
0.05986
0.02410
0.02464
0.03449
0.04442
0.02213
0.02760
0.02953
0.03974
0.02347
0.02884
0.02844
0.04401
1
1.02
1.48
1.56
1
1.22
1.34
1.64
1
1.25
1.19
1.65
1
1.02
1.43
1.84
1
1.25
1.33
1.80
1
1.23
1.21
1.87
1
0.89
1.04
1.00
1
1.04
0.88
0.91
1
0.98
0.88
1.00
1
0.82
0.91
1.03
1
1.00
0.98
1
1.00
1.03
1
1.16
1.42
1.56
1
1.18
1.52
1.80
1
1.27
1.35
1.65
1
1.24
1.58
1.79
1
1.25
1.83
1
1.23
1.81
-22.3
-29.5
-22.9
-28.8
-29.0
-28.9
Similar to Ea, ∆V‡ can also be calculated incrementally to evaluate its dependence on the
degree of hydration. The results are shown in Figure 4.13. The large scatter at very early ages (α
< 0.03) may be attributed to the fact that the early hydration rates do not follow the chemical
kinetics theory described by Eq. (4.2). However, it is also possible that such noncompliance is
due to the inevitable errors in the estimated “same degree of hydration” since the hydration rate
changes very rapidly with degree of hydration during very early periods. The activation volumes
appear to remain relatively constant until the degree of hydration exceeds 0.2. The variations in
86
activation volume may be explained by the fact that the temperatures of the samples do not
remain exactly constant as assumed and that different phases of cement probably hydrate at
different rates with slightly different activation volumes.
0
Class A
Class C
Class G
-10
Activation volume (cm3/mol)
Activation volume (cm3/mol)
0
-20
-30
-40
-50
0
0.2
0.4
0.6
Degree of hydration
0.8
Class H-P
Class H-I
Class H-II
-10
-20
-30
-40
-50
0
0.2
0.4
Degree of hydration
0.6
Figure 4.13: Variation of activation volume with degree of hydration
4.6 Verification of the Proposed Model
It was shown in section 4.2.2 that the normalized differential equation curve of a given
cement paste remains approximately invariant for different curing conditions. Therefore, the
hydration kinetics curves of any curing condition can be predicted by applying a scale factor C to
the experimental curves of a reference curing condition according to Eqs. (4.8) - (4.14). The
value of the scale factor depends on the curing temperature and pressure, which can be estimated
with Eq. (4.7). Table 4.3 summarizes how the scale factor can be used to transform the hydration
kinetics curves of a reference curing condition to predict those of any curing condition.
87
Table 4.3: Summary of the universal model to predict the hydration kinetics curves
Type of kinetics
curve
Reference
condition
Condition
to be predicted
Differential
equation
dα
= z (α )
dt
α = f (t )
dα
= Cz (α )
dt
α = f ( Ct )
dα
= g (t )
dt
dα
= Cg ( Ct )
dt
Integral
Derivative
x-axis
α
t
t
y-axis
Transformation
role
dα
dt
α
x-axis divided by C
dα
dt
x-axis divided by C
y-axis times C
y-axis times C
Table 4.1 and Table 4.2 indicate that the effect of curing pressure on cement hydration
kinetics is much smaller than that of curing temperature. The scale factor on hydration rate due
to a pressure increase of 51 MPa ranges from 1.56 to 1.82 while that due to a temperature
increase of 15.6 °C is much larger than 2. It is also shown in the tables that to (i.e. the starting
point of the accelerating stage or the end of the induction period), though difficult to determine
accurately, is not significantly affected by curing conditions. However, the model shown in
Table 4.3 would have predicted that such characteristic times differ from each other by a factor
of C, which is apparently incorrect especially for higher values of C. Since the degrees of
hydration achieved at to for all curing conditions are negligible, the hydration kinetics curves
may be offset to this point for the purpose of verifying the model. Figure 4.14 shows that
hydration kinetics curves at higher curing temperature can be predicted from those at ambient
temperatures with remarkable accuracy. Note that the scale factors used to make the predictions
were those obtained from Eq. (4.16) shown in Table 4.1. Small offsets were sometimes observed
between experimental and predicted curves primarily due to difficulties in accurately
determining to. The nearly perfect agreements between measured and predicted hydration
kinetics for all different types of cement also indirectly support the proposed linear reduction rate
of CS0 with increasing temperature (see Section 3.6) because measured degrees of hydration are
strongly dependent on the estimated values of CS0.
88
Measured
Predicted
1
0.6
60 °C
0.4
40.6 °C
0.2
0
Class A
0
20
40
Time (h)
24.4 °C
Degree of hydration
Degree of hydration
0.8
60 °C
40.6 °C
0.4
Class C
0.2
0
20
40
Time (h)
26.9 °C
60
0.8
0.6
60 °C
0.4
40.6 °C
0.2
Class G
0
20
40
Time (h)
25 °C
60
Degree of hydration
Degree of hydration
0.6
0
60
0.8
0
0.8
0.6
0.4
60 °C
40.6 °C
0.2
0
Class H-P
0
20
40
Time (h)
25.6 °C
60
Figure 4.14: Measured and predicted hydration kinetics of different types of cement cured at
different temperatures (Ambient condition as the reference)
As to the effect of curing pressure, it was not necessary to offset the hydration kinetics
curves, because the scale factors were so small. Figure 4.15 shows that the hydration kinetics
curves obtained experimentally at the curing pressure of 51.7 MPa agree nearly perfectly with
those predicted using experimental data obtained at the reference curing pressure of 0.69 MPa.
Figure 4.16 further demonstrates that even the details (such as double peaks) of the derivative
curves are accurately predicted with the proposed model.
89
1
0.6
Class C
0.5
Degree of hydration
Degree of hydration
0.8
Class G
0.6
0.4
Class A
Class H-P
0.2
0
0.4
0.3
0.2
0
10
20
Time (h)
30
Class H-I
0.1
Measured
Predicted
0
Class H-P
40
Measured
Predicted
0
10
20
Time (h)
30
40
Figure 4.15: Measured and predicted hydration kinetics of different types of cement cured at 51.7
MPa (0.69 MPa curing pressure as the reference)
Measured
Rate of hydration (/h)
0.06
0.04
Class A
Rate of hydration (/h)
Class C
0.05
0.02
0
0
10
20
30
40
0.06
0
0
10
20
30
40
0.04
0.04
Class H-P
Class G
0.02
0.02
0
Rate of hydration (/h)
Predicted
0.1
0
10
20
30
40
0.04
0
0
10
30
40
0.04
Class H-II
Class H-I
0.02
0
20
0.02
0
10
20
Time (h)
30
40
0
0
10
20
Time (h)
30
40
Figure 4.16: Measured and predicted hydration kinetics of different types of cement cured at 51.7
MPa (0.69 MPa curing pressure as the reference)
90
4.7 Summary
Cement chemical shrinkage, which can be easily measured at both different temperatures
and pressures, is an important alternative to the isothermal calorimetry method in evaluating
cement hydration kinetics. Total chemical shrinkage is approximately proportional to degree of
hydration, with a proportionality constant decreasing with both curing temperature and curing
pressure. A one-parameter model is developed in this chapter to model the effect of curing
temperature and pressure on cement hydration kinetics. For the same cement paste cured at
different temperatures and pressures (constant with time), the hydration kinetics curves differ
from each other by only a factor of C. At any degree of hydration, when the rate of hydration is
multiplied by C due to a change in temperature and/or pressure, the time to reach this particular
degree of hydration is multiplied by 1/C. Therefore, the kinetics curve at any curing temperature
or pressure can be accurately predicted by simply rescaling a kinetics curve obtained at a
reference condition. Some variability is found during very early periods of hydration (initial
reaction period and induction period). Therefore, it is sometimes necessary to offset the zero time
and degree of hydration to the end of the induction period, especially for larger values of C. The
scale factor is related to the activation energy and the activation volume of the cement by Eq.
(4.7). The activation energies (Ea) of Class A, C, G, H-P, and H-II cements used in this study
were found to be 52.6, 48.8, 50, 42.5 and 44.3 kJ/mol, respectively, while the activation volumes
(∆V‡) of these cements were found to be -22.3, -29.5, -22.9, -28.8 and -28.9 cm3/mol,
respectively. These values are only approximate due to limitations of the temperature control
scheme of the test apparatus. More accurate values of activation energies are presented in
Chapter 6.
91
CHAPTER 5 : NUMERICAL MODELING OF CEMENT
HYDRATION KINETICS
5.1 Introduction
It has been shown in Chapter 4 that the effect of curing temperature and pressure (external
factors) on cement hydration can be modeled without knowing the particular hydration
mechanisms since they appear to only have kinetics effects on the hydration reactions. For the
model developed in Chapter 4 (Table 4.3), the advantage is that it is applicable to any type of
cement while the disadvantage is that its application requires an experimental hydration kinetics
curve. In order to model the hydration process purely theoretically and to gain further insight into
the role of the various influencing factors, it is necessary to further investigate the detailed
hydration mechanisms. The different clinker phases of Portland cement are known to react at
different rates and possibly interact with each other, which are the primary causes of the
complexity of the hydration process. As the main composition, pure C3S or alite (an impure form
of C3S) has very similar hydration kinetics as Portland cement itself (Taylor 1997a). In this
chapter, a particle-based numerical model is developed based on a simplified C3S hydration
mechanism (i.e. hydration in stirred dilute suspensions with constant lime concentration). The
model is directly applied to C3S hydration data published in the literature to study the effect of
particle size and quantity of initial nuclei on hydration. It is slightly modified and applied to
Class H (H-II and H-P) cement hydration data obtained in this study to investigate the effect of
cement composition, w/c ratio, and curing conditions on hydration.
C3S is known to exhibit seven polymorphs (three monoclinic, three triclinic, and one
trigonal) and is present in Portland cement in various forms through ion-stabilization (Bigare
92
1967, Maki 1989, Taylor 1997a, Hewlett 1998, Costoya 2008). Although it has been shown that
the reactivity of these different forms may be different, their general reaction mechanisms appear
to be similar (Peterson 2006a). Hydration of C3S produces calcium-silicate-hydrate (C-S-H) gel
and calcium hydroxide, which (under saturated condition) can be approximated as:
3CaO ⋅ SiO 2 + ( 7 - x ) H 2 O = xCaO − SiO 2 − 4H 2 O + (3 − x)Ca(OH) 2
(5.1)
where x is the CaO to SiO2 ratio (C/S) of the C-S-H formed, which varies with the calcium
hydroxide concentration in the solution (Nonat 1997, Gartner 2002). The reaction takes place via
a dissolution-precipitation process. More specifically, Eq. (5.1) can be divided into the following
three steps:
Ca 3SiO5 + 3H 2 O → 3Ca 2 + + 4OH − + H 2SiO 4 2−
(5.2)
xCa 2 + + 2(x − 1)OH − + H 2SiO 4 2− + (4 − x)H 2 O → xCaO − SiO 2 − 4H 2 O
(5.3)
(3 − x)Ca 2 + + 2(3 − x)OH − → (3 − x)Ca ( OH )2
(5.4)
Due to the exothermic nature of the reaction, hydration kinetics of C3S is most frequently
measured by monitoring the rate of heat evolution with isothermal calorimetry. It should be
noted though that the exothermic reaction step is the dissolution of C3S (Eq. (5.2)), rather than
the formation of hydration product (Eqs. (5.3) and (5.4)) (Grant 2006). Measurement of ion
concentrations during C3S hydration in suspensions or paste showed a consistent rise in calcium
concentration and a consistent decrease of silica concentration (after a peak reached within a few
minutes of hydration) until the end of the induction period, which means that the three steps do
not progress at the same pace during the very early stage (typically t < 2h) (Brown 1984,
Grutzeck 1987). The ion concentration remains relatively stable afterwards, suggesting a balance
is reached between dissolution and precipitation.
93
Similar to Portland cement, the hydration process of C3S can be classified into the same
five periods as shown in Figure4.1. The early hydration mechanisms (transition from period 1 to
period 2) are still not clearly understood today. One hypothesis is that a meta-stable layer of
hydration product formed at the end of period 1 creates a diffusion barrier to dissolving ions
(Jennings 1986a, Gartner 1989, Taylor 1997a, Gartner 2002). Another hypothesis attributes the
slow reaction during period 2 to the difficulty in nucleating hydration products (Garrault 2001,
Gartner 2002, Garrault 2006). Due to these uncertainties and the small total degree of hydration
achieved, the first two periods are typically ignored in modeling hydration. The last three periods
are commonly believed to be controlled by a mechanism that gradually transforms from
nucleation and growth (NG) controlled to diffusion controlled (DC). It is now well established
that period 3 is controlled by the NG mechanism (Thomas 2011). However, many details regard
to the shift of the rate-controlling mechanism are still uncertain and there is no clear separation
point between period 4 and period 5. Some recent studies have shown that the NG mechanism
can be used to model period 4 but did not rule out an eventual transition to the DC mechanism
(Thomas 2007, Thomas 2011, Bishnoi 2009a, Kumar 2011). Bishnoi and Scrivener (2009a)
argue that the DC mechanism cannot explain period 4 unless drastic changes in the transport
properties of C-S-H from different alite particles are assumed, which is found to be not true in
this study. It is possible to model Period 4 by assuming that a gradual transition from the NG
mechanism to the DC mechanism occurs during this period. The transport properties of C-S-H
do not need to be assumed to vary with particle size to generate a good fit to test data.
Since cement hydration will probably eventually become diffusion controlled, none of the
models developed based on the NG mechanism can be used to explain hydration at later ages
(>24 h). A diffusion model, which cannot explain hydration at early ages (typically t < 10-20 h),
94
has been successfully used to model cement hydration up to 1000 hours (Berliner 1998, Ridi
2003). Therefore, in order to successfully model the entire hydration process, it is most
appropriate to combine a NG mechanism (for early stage hydration) with a DC mechanism (for
later stage hydration). The effectiveness of such a combined model in simulating the hydration
behavior of C3S paste has been demonstrated by many investigators (FitzGerald 1998, Berliner
1998, FitzGerald 1999, FitzGerald 2002, Damasceni 2002, Ridi 2003, Allen 2004, Peterson 2005,
Peterson 2006a, Peterson 2006b, Peterson 2009). However, since many details with regard to the
shift of the rate controlling mechanism are still not well established today, test data were fitted
with two completely different models in these studies, which often resulted in discontinuities at
the transition points. The estimated time of transition in these studies typically ranged from 10 h
to 20 h for tests conducted at 20 °C. In this study, a new NG model developed from the particle
level is proposed while a previously proposed DC model on the same scale is modified such that
the two models can be continuously connected. As will be shown later, combining of the two
models can smoothly simulate cement hydration kinetics and fits experimental data almost
perfectly.
How the nuclei are formed during C3S hydration is probably the most controversial issue in
developing NG models. It has been suggested that the decrease of silicon ion concentration
during early hydration (from the peak reached within a few minutes after mixing to the end of
the induction period) correlates with the precipitation of C-S-H nuclei from the solution
(Garrault-Gauffinet 1999, Garrault 2001, Garrault 2005, Garrault 2006). Due to the relatively
short duration, this period of hydration may be simplified as an instantaneous nucleation process.
The “growth” of these initially formed nuclei (fixed number) then results in the acceleration
period of C3S hydration. Thomas et al. (2009) pointed out that the macroscopic growth kinetics
95
is technically a nucleation process in that the formation of new C-S-H particles is caused by
stimulation of existing C-S-H particles. Such autocatalytic behavior may imply that nucleation at
new sites is less likely to happen during the “growth period” as it is easier to nucleate on existing
C-S-H surfaces.
5.2 Theoretical Background
An appropriate simulation of cement hydration is essential for predicting and optimizing
the various physical and mechanical properties of cement-based materials. As pointed out by
Thomas et al. (2011), when different “mathematical constructs” are compared, it is useful to
draw a distinction between “models” and “simulations”. The goal of the model proposed here is
to reproduce the hydration kinetics curves only. Therefore, it shall be distinguished from the
more complex “simulations” presented in other studies (Jennings 1986b, Bentz 1997, van
Breugel 1995a, van Breugel 1995b, Bullard 2007a, Bullard 2007b, Bishnoi 2009a, Bishnoi
2009b). However, since the model proposed here is particle based, it has the potential to be
further developed to simulate the development of the microstructure and other properties of
cement-based materials.
Recent reviews of the numerous mathematical models and simulations developed in the
past 40 years have shown that a complete simulation that can accurately simulate both the
hydration kinetics and the microstructure development still does not exist (Xie 2011, Thomas
2011). Xie and Biernacki (2011) fitted a number of different models to the hydration kinetics
data of C3S paste at early ages and found only a few models can provide reasonable fits to both
the integral and the derivative curves of hydration kinetics. These include the Pommersheim et al.
model (1982), the boundary nucleation and growth (BNG) model (Cahn 1956, Thomas 2007),
96
and the Bishnoi and Scrivener model (2009a). It appears that the Pommersheim et al. model
starts to deviate dramatically from experimental data at about 15 hours (Xie 2011). Thomas has
shown that the BNG model starts to deviate from experimental data at about 10 to 30 hours
depending on the curing temperature (Thomas 2007). Although Zhang et al. (2010) have found
that the BNG model can provide an excellent fit to the experimental data of Class H cement
paste hydration up to 70 hours, it is not clear how the goodness of fit varies with w/c ratio and
curing condition as only one representative fit was presented. In addition, some model
parameters of the BNG model do not comply with the Arrhenius equation (Thomas 2007), which
is typically used to describe the temperature dependence of the reaction rate constant. The
Bishnoi and Scrivener model has been shown to provide an excellent fit to C3S paste hydration
up to 24 hours (Bishnoi 2009a, Kumar 2011). However, the success of this model requires an
assumption that the bulk density of C-S-H varies significantly with time (from 0.25 to 1.9 g/cm-3)
(Kumar 2011), which is not substantiated experimentally. In addition, it is not clear how the
goodness of fit and model parameters vary with w/c ratio and curing condition as only the effect
of particle size distribution was investigated.
More details about the limitations of the existing models in predicting cement hydration
kinetics has been discussed thoroughly in the recent review papers (Xie 2011, Thomas 2011) and
will not be repeated here. Instead, we will only emphasize the important theories and
formulations that will be further used in developing our own model. The essence of the classical
equations for modeling the nucleation and growth mechanism (Avrami 1939, Avrami 1940) is
the introduction of the concept of extended volume to account for impingement between different
growing nuclei. This concept, together with a similar one (extended area), has been used
repeatedly to modify the original NG equations such that they can be applied more appropriately
97
to cement hydration (Cahn 1956, Thomas 2007, Bishnoi 2009a). The classical NG equations
were originally developed for solid phase transformations from one phase (α) to another (β). The
extended volume (Vβe) of phase β, is defined as the total volume of the phase assuming no
overlaps between growing nuclei. The real volume increase of phase β during each time
increment is believed to be proportional to the volume fraction of untransformed α, i.e.,
 V − Vβ 
V 
dVβ = dVβe  α  = dVβe 

V 
 V 
(5.5)
where V is the total volume, while Vα and Vβ are the volumes of phase α and β, respectively.
One of the most effective and straightforward models for the DC mechanism was proposed
by Fujii and Kondo (1974). The model has been applied successfully to fit experimental data of
C3S hydration by many investigators (FitzGerald 1998, Berliner 1998, FitzGerald 1999,
FitzGerald 2002, Damasceni 2002, Ridi 2003, Allen 2004, Peterson 2005, Peterson 2006a,
Peterson 2006b, Peterson 2009). In this model, it was assumed that the hydration products
formed during the propagative NG stage do not hinder the diffusion of water due to their
relatively high permeability. It was further assumed that after the C3S particle is completely
covered by hydration product, a less permeable layer is formed between the earlier hydration
products and the anhydrous C3S core. These assumptions are consistent with microstructure
observations, as it has been found that C-S-H formed at early stages (outer product) had foilshape or fibrillar morphology and low density while the inner C-S-H formed at later stages was
much denser and was composed of small globular particles more homogeneously distributed
(Costoya 2008). Therefore, the rate of reaction after the NG stage was assumed to be controlled
by diffusion of water through the inner hydrated layer. Consequently, if hydration enters the DC
stage at time td, then (Fujii 1974),
98
dR
D
=−
dt
R − Rd
(5.6)
Rd − R = 2D(t − td )
(5.7)
where R and Rd are the radii of the anhydrous core after hydration time t and td, respectively, and
D is the diffusion constant. However, it is apparent that Fujii’s model does not apply to the
vicinity of td (i.e. the transition from the NG to DC stage) as it predicts the hydration rate to
approach infinity (Eq. (5.6)). As will be discussed later, the problem may be resolved by
introducing a pseudo time when hydration become diffusion controlled, which precedes td.
It is interesting to note that Eq. (5.7) has a similar form as the equation of the penetration
depth (a measure of the distance travelled by an average diffusing atom) derived from diffusion
theory (Christian 2002),
d = 2 Dt
(5.8)
where d is the penetration depth at time t.
5.3. Model Formulation
Since cement hydration is an extremely complex process, it is critical to use simplified
processes to study the hydration mechanisms. For example, C3S hydration is simpler than
Portland cement hydration because there are no different clinker phases; C3S hydration in stirred
dilute suspensions is simpler than C3S paste hydration because there are no interactions between
different particles. Garrault et al. (Garrault-Gauffinet 1999, Garrault 2001, Garrault 2005,
Garrault 2006) performed a series of experimental studies using probably the simplest process
one can achieve: C3S hydration in stirred dilute suspensions with constant lime concentrations. In
these studies, impingement between different nuclei are only limited to C-S-H since it is not
possible to nucleate calcium hydroxide (CH). The liquid-to-solid ratio used was 50:1. The lime
99
concentration was controlled by maintaining a constant value of electrical conductivity of the
solution (a small amount of solution was replaced with pure water whenever there was an
increase in electrical conductivity). The mathematical equations of our model are formulated
particularly to fit the test results of these experimental studies. The original goal of this model
was to derive some critical physical parameters of C3S hydration in dilute suspensions, such as
nuclei growth rate, number of nuclei per unit area, and diffusion constant, so that they can be
compared with those of other studies. However, as will be shown in Section 5.5, the model can
also be applied to cement paste hydration with just some minor modifications.
The dependence of hydration rate on particle sizes during both the NG and DC stages has
been widely recognized (Allen 2004, Garrault 2006, Costoya 2008, Bentz 2010). Studies have
also found that the effect of particle size distribution on overall hydration kinetics can be
accounted for by a simple law of mixtures (linear addition) (Costoya 2008, Bentz 2010).
Therefore, the best way to model hydration kinetics using Eq. (5.7) would be to apply the
equation to individual particles of given size and then apply the law of mixtures. This allows to
identify individual NG to DC transition times for each particle size instead of a bulk transition
time for the entire representative volume. Such implementation calls for a NG model at the
particle scale as well, which will be discussed in Section 5.3.1.
5.3.1 Modeling the nucleation and growth controlled stage
Experimental results have shown that during C3S paste hydration C-S-H gel primarily
nucleates and grows on particle surfaces while calcium hydroxide (CH) typically nucleates and
grows in the pore spaces, with most researchers believing the former to be rate controlling
(Gartner 2002, Costoya 2008). The fact that C3S hydration in solutions of constant lime
100
concentrations (with no nucleation of CH) exhibits a similar behavior as C3S paste hydration
further confirms this assertion (Garrault 2001). In the model developed here it is assumed that all
nuclei are formed instantaneously on particle surfaces at the beginning of hydration. These nuclei
then grow uniformly, resulting in reduction of the anhydrous core due to dissolution. Figure 5.1
shows a schematic of such mechanism.
Figure 5.1: Schematic of assumed C3S hydration mechanism during NG stage
Let A0 be the original surface area of the C3S particle, and n the number of C-S-H nuclei
nucleated per unit surface area at the beginning of hydration. Assume all nuclei grow at two
uniform rates: parallel to the particle surface at a rate of g1 and normal to the particle surface at a
rate of g2. In this study, the extended area is defined as the total particle surface area that would
be covered by hydration product if no impingement between different nuclei occurred. Under the
assumption that the size of a nucleus is much smaller than that of a C3S particle, the total
extended area of the particle surface covered by hydration product at time t may be approximated
by,
Ahe = A0 n ⋅ π ( g1t )
2
(5.9)
101
The increase of the extended area during each time increment is derived by differentiating Eq.
(5.9),
dAhe = 2π A0 ng12tdt
(5.10)
Only a fraction of this area increment is real, the portion that lies on previously formed hydration
product is virtual (i.e. does not exist). Since the distribution of nuclei is random, the fraction of
the extended area that forms during each time increment that is real will be proportional to the
area fraction of uncovered particle surface, i.e.,
 A − Ah
dAh = dAhe 
 A
Ah 

2 
 = 2π A0 ng1 t 1 −  dt
A


(5.11)
where Ah is the surface area of the particle covered by hydration product while A is the total
surface area of the anhydrous core, which changes with time due to dissolution.
Assume the original radius of the spherical anhydrous particle is R0, and the radius of the
anhydrous core at time t is R (Figure 5.1), then the relationship between degree of hydration (α)
and R can be obtained as follows,
R
V
R3 − R 3
α = hc = 0 3 = 1 −  
V0
R0
 R0 
3
(5.12)
where Vhc and V0 are the hydrated volume and the total original volume of the particle,
respectively. The above equation can be written as,
R = R0 (1 − α )
1/3
(5.13)
Therefore, the size of the anhydrous core is a function of degree of hydration. Eq. (5.11)
becomes,


A
dAh = A0 ng12t  2π − 2 h 2/3  dt

2 R0 (1 − α ) 

(5.14)
102
As will be shown later, the parameters n and g1 cannot be determined independently when fitting
the proposed model to experimental data. A parallel growth rate constant S = ng12, which
describes how fast the nuclei are spreading around the surface of the particle, is introduced here
to couple the two parameters. After substitutions, Eq. (5.14) becomes,


Ah
 dt
dAh = 2π St  4π R02 −
2/3


α
1
−
(
)


(5.15)
During each time increment, the covered surface would grow a thickness of g2dt, hence the
incremental change of the total volume of hydration product can be expressed as,
1
dVh = Ah g 2 dt + dAh g 2 dt
2
(5.16)
Apparently, the second term on the right-hand side of the above equation can be ignored since it
is a higher order infinitesimal quantity. Therefore,
dVh = Ah g2 dt
(5.17)
If we assume the hydration of 1 unit volume of C3S produces volume c of C-S-H hydration
product, then,
Vh = cV0α
(5.18)
Ah g 2
AK
dt = h dt
cV0
V0
(5.19)
Therefore,
dα =
Again, the parameters g2 and c cannot be determined independently when fitting the proposed
model to experimental data. A consumption rate constant K = g2/c, which describes how fast the
particle is being consumed in the direction perpendicular to its surface, is introduced here to
couple the two parameters. The degree of hydration as a function of t can be obtained by
numerically integrating Eqs. (5.15) and (5.19). The equations are only valid for the NG stage, i.e.
103
before the particle is completely covered by C-S-H. Therefore the radius of the anhydrous core
(R) shall be calculated continuously to make sure the model is not applied beyond its applicable
condition (Ah < 4πR2).
5.3.2 Modeling diffusion controlled stage
As mentioned earlier, Fujii’s DC model (Eq. (5.7)) has to be modified as follows to
eliminate the inaccurate portion of the curve in the vicinity of td,
R pd − R = 2 D ( t − t pd )
(5.20)
where tpd is the pseudo time when hydration becomes diffusion controlled and Rpd = R(tpd) is the
pseudo radius of the anhydrous core at time tpd. One crucial boundary condition for the transition
from nucleation and growth controlled hydration to diffusion controlled hydration is that the rate
of hydration should be continuous. In other words, the rate of hydration must be the same at the
transition point. As shown in Eq. (5.13), the degree of hydration is directly related to the radius
of the anhydrous core of the particle. The latter will be used for the following derivations,
because its rate of change is much easier to calculate. For diffusion controlled hydration,
dR
D
=−
dt
2 D ( t − t pd )
(5.21)
For nucleation and growth controlled hydration (derived from Eqs. (5.13) and (5.19)),
A
dR
=− h 2 K
dt
4π R
(5.22)
When Ah = 4πR2 (meaning that the anhydrous core is completely covered by hydration product),
the right hand side of Eq. (5.22) reduces to K, which is simply the consumption rate constant.
104
We assume that the transition from the NG stage to the DC stage happens at the time when
the anhydrous core is completely covered by hydration product. The difference between the
pseudo time (tpd) and the real time (td) when hydration enters the DC stage can be derived by
combining Eqs. (5.21) and (5.22),
D
2K 2
(5.23)
Rpd = Rd + 2 D ⋅ lag
(5.24)
lag = td − t pd =
From Eq. (5.20),
td and Rd are the values of t and R obtained at the end of the NG stage (Ah = 4πR2). Combining
Eqs. (5.13) and (5.21) and replacing the pseudo terms with the real ones, the equation for the DC
stage is derived as follows,
α = 1−
(
1
Rd + 2 D ⋅ lag − 2 D ( t − td + lag )
R03
)
3
(5.25)
5.3.3 Modeling the total hydration kinetics
Based on the previous analyses, the basic simplifying assumptions of the proposed model
can be summarized as follows.
1. All cement particles are spherical and hydrate individually (interactions between particles at
later stages are indirectly accounted for by their effect on the diffusion constant).
2. Nucleation of hydration products is site-saturated (fixed number of nuclei) and the growth of
those nuclei formed on the surface of the cement particles is the rate controlling mechanism
during the NG stage.
3. All nuclei grow at two constant rates, parallel and perpendicular to the particle surface.
105
4. Hydration of each cement particle enters the DC stage as soon as its surface is completely
covered by hydration products.
5. Only inner hydration products (i.e. those formed in the space between the anhydrous cement
particle and the hydration products formed during the NG stage) act as diffusion barrier
throughout the DC stage of hydration (Fujii 1974).
With these assumptions, the hydration of a single cement particle with a radius of R0 can now be
modeled continuously with Eqs. (5.15) and (5.19) for the NG stage (Ah < 4πR2) and Eq. (5.25)
for the DC stage with rate-controlling parameters: S, K, and D. A flow chart showing how these
equations can be implemented in a computer program is presented in Figure 5.2. The total degree
of hydration of a cement sample is then simply the weighted average degree of hydration of all
particles in the sample,
N
α T ( t ) = ∑ α ( R0 , t ) ⋅ f ( R0 )
(5.26)
1
where αT(t) is the total degree of hydration of the sample; α(R0,t) is the degree of hydration of
particles with a mean radius of R0; f(R0) is the weight fraction of particles with a mean radius of
R0 and N is the total number of gradations according to the particle size distribution (PSD). Eq.
(5.26) can be easily implemented by a matrix multiplication. The final computer program takes
particle size distribution data as input and gives the degree of hydration as a function of time as
output. Fitting the model with experimental data can be performed by manually adjusting the
three parameters to achieve the best agreement.
106
Figure 5.2: The flowchart of computer simulation of the hydration of a single C3S particle
5.4 Model Application: C3S Hydration in Dilute Suspensions
5.4.1 Effects of particle size distribution
Garrault et al. (2006) studied the effect of particle size on hydration kinetics of C3S in
stirred saturated lime solutions with a liquid-to-solid ratio of 50:1. The lime content in the
solution was kept constant to avoid dependence of hydration rate on lime concentration. A total
of five samples with median diameters of 7, 10, 11, 12.5, and 14 µm, respectively, were obtained
107
by sedimentation of a C3S suspension in ethanol. The total amount of hydration product (C-S-H)
produced was measured for a period of 20 hours for each sample. The data was re-digitized and
converted to degree of hydration to be fitted with the proposed model. As shown in Figure 5.3,
excellent agreement was obtained between experimental results and the model. The fitted
parameters are presented in Table 5.1. The parallel growth rate constant (S = ng12) was found to
increase with particle size. Since the nuclei parallel growth rate (g1) is not likely to be dependent
on particle size, the result suggests that the initial number of nuclei nucleated per unit surface
area (n) increases with particle size (or decreases with decreasing particle size). In other words,
the total initial number of nuclei precipitated of a sample does not increase as fast as specific
surface area. This is supported experimentally, because the cumulative heat release (proportional
to the degree of reaction, and hence initial number of nuclei precipitated) at the end of the
induction period was found to increase at a slower pace than specific surface area (Costoya
2008). However, a more accurate relationship between initial number of nuclei precipitated and
particle size would have to be established by further experimental work. The random variations
of K with respect to particle size may be attributed to experimental errors, especially in particle
size distribution measurements. For example, samples 2 and 3 had very similar particle size
distributions (Figure 5.4), but their degree of hydration curves were quite different (Figure 5.3).
The author also pointed out that “Each size distribution curve is wide and can be indicative of the
presence of agglomerates formed by elementary particles.” It is not clear what step size was used
in the original study for measuring particle size distribution. In the present study more than 50
data points (gradations) were used to re-digitize each cumulative particle size distribution curve.
108
0.4
Experimental Results
Modeled Results
0.35
1
2
Degree of Hydration
0.3
3
0.25
4
0.2
5
0.15
0.1
0.05
0
0
200
400
600
800
Time (min)
1000
1200
1400
Figure 5.3: Experimental (Garrault 2006) and modeled results of degree of hydration of samples
with different particle sizes
Table 5.1: Model Parameters for different particle size brackets
Sample No.
1
2
3
4
5
Median Particle Diameter (µm)
7
10
11
12.5
14
a
2
Specific Surface Area (cm /g)
2730
1911
1737
1529
1365
Specific Surface Areab (cm2/g)
3212
2340
2245
1955
1531
0.0112 0.0115 0.0162 0.0166 0.0180
S=ng12 (h-2)
Model
K=g2/c (µm/h)
0.062 0.075 0.057 0.050 0.055
Parameters
2
D (µm /h)
0.0030 0.0021 0.0018 0.0018 0.0018
a
: Determined from original PSD data by original author (Garrault 2006)
b
: Determined from reproduced data by re-digitizing the PSD graph (Figure 5.4)
109
Figure 5.4: Cumulative particle size distribution curves for the five samples (Garrault 2006)
The particle size distribution is an implicit influence factor of the proposed model. Figure
5.5 shows the effect of initial particle size on hydration kinetics using the model parameters
obtained for sample #3. The transition point from the NG to the DC stage was found to occur
earlier for smaller particles than larger ones with the degree of hydration achieved at the
transition point decreasing significantly with increasing particle size. This is consistent with
microscopic observations (Costoya 2008) of alite and C3S paste hydration, where it has been
found that smaller particles are covered sooner than larger ones. Therefore, for a sample with
multiple particle sizes, the transition of the rate controlling mechanism occurs through a period
of time instead of a single point. As shown in Figure 5.6, the peak hydration rate of a sample is
always achieved during the transition period, with its relative position within the period
depending on the particle size distribution of the sample. C3S paste hydration (Costoya 2008)
appears to reach its peak rate slightly later than hydration in constant lime solutions.
110
1
R =0.5
0
0.9
R =1
0
0.8
R =2
0.7
R =4
Degree of Hydration
0
0
R =8
0.6
0
R =16
0
0.5
R =32
0
0.4
R =64
0
0.3
0.2
0.1
0
0
200
400
600
Time (min)
800
1000
Figure 5.5: Effect of initial particle size on hydration kinetics
(R0 is the initial particle radius in µm; ◊ indicate transition points)
-4
8
x 10
Sample #1
Sample #2
Sample #3
Sample #4
Sample #5
Rate of hydration (/min)
7
6
5
4
3
2
1
0
0
200
400
600
Time (min)
800
1000
1200
Figure 5.6: Modeled rate of hydration of samples with different particle sizes
(● indicate the transition period from NG to DC stage)
1200
111
The original Fujii’s DC model (obtainable by combining Eqs. (5.7) and (5.13)) was
typically used for the derivation of D in earlier studies,
(1 − α )
1/3
= (1 − α d )
1/3
− ( 2D )
1/ 2
R0−1 ( t − td )
1/ 2
(5.27)
where αd is the degree of hydration at td, the time when hydration becomes diffusion controlled.
The equation was apparently developed based on a single particle size, but it is nevertheless used
to model hydration of C3S samples with a broad range of particle sizes, which raises the question
of how R0-1 should be determined. The relatively large variation in previously reported values of
D can be largely attributed to inconsistent methods of estimating R0-1: Some authors used the
weighted mean of the inverse of the radius from the PSD data (Allen 2004); some took the
inverse of the weighted mean radius from the PSD data (Damasceni 2002, Peterson 2005,
Peterson 2006b, Peterson 2009); some took the inverse of the mean radius calculated from
specific surface area (Fujii 1974); still most others did not clearly specify how they obtained the
value (FitzGerald 1998, FitzGerald 1999, FitzGerald 2002, Ridi 2003). It has been shown that
the values obtained by these different methods can vary dramatically (Berliner 1998, Allen 2004),
depending on the particle size distribution of the sample. In the study performed by Berliner et al.
(1998), it is likely that the inverse of the mean diameter instead of the radius was used. The
mistake could be a result of interpretation of the PSD, in which particle size is expressed as
diameter (not radius). The claimed mean radius (18.2 µm) corresponds to approximately 85
percentile on the cumulative particle size distribution curve, which is unreasonably high. A mean
diameter of 18.2 µm (which corresponds to approximately 70 percentile) is more reasonable.
This could have resulted in the values of D obtained to be four times as high as it should be.
The fact that C3S hydration has been successfully modeled with Eq. (5.27) in previous
studies suggests that every sample has a characteristic particle size that can be used to model its
112
total hydration kinetics (rather than taking the weighted average of different size particles), at
least during the DC stage. The characteristic sizes for the five samples were found by manual
fitting to be 3.18, 4.3, 4.5, 5.1, and 6.5 µm respectively. The reciprocals of these values are very
close to the weighted mean inverse radius from the PSD data of each sample. The results
modeled with these characteristic particle sizes were compared with those modeled with multiple
sizes (the entire particle population of the samples) in Figure 5.7. The curves obtained with these
two different approaches deviated slightly from each other during the NG stage, but were almost
identical during the DC stage. As will be discussed later, the capability of our model to
reproduce the total degree of hydration of a sample using a single particle size allows it to be
fitted with hydration data where particle size distribution data is not available.
0.4
Multiple sizes
Single size
0.35
Degree of Hydration
0.3
0.25
0.2
0.15
0.1
0.05
0
0
200
400
600
Time (min)
800
1000
1200
Figure 5.7: Comparison between modeled curves obtained with single particle size vs. multiple
particle sizes (◊ indicate transition points for single-particle model while ● indicate the transition
period for multiple-particle model)
113
5.4.2 Physical meanings of model parameters
The nuclei growth rates on a polished C3S (sintered pellet) surface has been estimated
experimentally using Atomic Force Microscope without lime concentration control [25]. The
estimated results were 4.1 x 10-11 m/s (0.148 µm/h) for parallel growth rate (g1) and 1.8 x10-11
m/s (0.065 µm/h) for perpendicular growth rate (g2). Substituting the estimated value of g1 into
the fitted values of S in Table 5.1, the number of nuclei per unit surface area was estimated to
range from 0.5 to 0.8 µm-2. For comparison, Scherer et al. (2011) estimated a value of 6 µm-2 by
applying the boundary nucleation model (assuming a fixed number of nuclei) to fit the chemical
shrinkage data of Portland cement. Bullard (2011) estimated a value of 1.42 µm-2 for C3S
hydration using the kinetic cellular automaton model (Bullard 2008, Bullard 2010). Assuming a
density of 3.15 cm3/g for C3S and 2.0 cm3/g for C-S-H, Bishnoi and Scrivener (2009a)
determined that the hydration of 1 unit volume of C3S would generate 1.569 unit volumes of CS-H hydration product. Therefore, the value of c may be assumed to be 1.569 in this study. Note
that there is no nucleation of CH due to lime concentration control. Substituting this value into
the fitted values of K in Table 5.1, the perpendicular growth rate was estimated to range from
0.08 to 0.12 µm/h, also in reasonable agreement with experimental observations (0.065 µm/h).
The fitted value of diffusion constant D (1.8 to 3 x 10-15 m2/h) is in good agreement with
previously reported values, which are typically of the order of 10-16 to 10-15 m2/h (FitzGerald
1998, Berliner 1998, FitzGerald 1999, FitzGerald 2002, Damasceni 2002, Ridi 2003, Allen 2004,
Peterson 2005, Peterson 2006b, Peterson 2009). Theoretically, when C3S hydrates in dilute
suspensions, the diffusion constant should only depend on the permeability of the diffusion
barrier (i.e. the inner C-S-H hydration product in this case). Therefore, D should be independent
of particle sizes. The fitted results generally confirmed the theory with the exception of Sample
114
#1, which was found to have a much higher (67%) value of D. This could be due to experimental
difficulties of accurately measuring the sizes of fine particles or due to the limitation of the
model when applied to fine particles since Sample #1 contained the largest amount of fine
particles. For C3S or cement paste hydration, due to impingement between different particles, the
effective contact area between capillary water and C3S particles will decrease as hydration
progresses. One would expect the “apparent” diffusion constant (i.e. the fitted value) to decrease
with time, although this decrease may be very slow. Similarly, since inter-particle spacing is
greatly affected by w/c ratio, one would also expect the apparent diffusion constant to decrease
with decreasing w/c ratio as a result of increasing impingement. This has been confirmed by
experimental studies (Berliner 1998). Therefore, the diffusion constant obtained from this study
should reflect the real permeability of the inner C-S-H hydration product as the particles are
entirely surrounded by water all the time. As will be shown in Sections 5.5.2 and 5.5.3, the fitted
values of diffusion constant D for Portland cement hydration at ambient temperatures are about
one order of magnitude higher than those obtained here. This is probably due to the presence of
other hydration products in the diffusion barrier, which increases its permeability.
By changing the value of each parameter of the model individually, their effects on the
total hydration kinetics can be observed. The modeled results of Sample #3 were used for such
demonstration (Figure 5.8). As shown in the figure, the parallel growth rate constant S mainly
controls the duration of the NG stage; the consumption rate constant K mainly controls the
degree of hydration achieved at the transition point between the NG and DC stages; while the
diffusion constant D controls the rate of reaction during the DC stage.
115
0.35
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
S=0.0081
Degree of Hydration
0.3
S=0.0162
S=0.0324
0.25
0.05
D=0.0009
D=0.0018
D=0.0036
D=0.0072
0.1
K=0.029
K=0.057
0.05
0.05
K=0.115
0
0
400
800
Time (min)
1200
0
0
400
800
Time (min)
1200
0
0
400
800
Time (min)
1200
Figure 5.8: Effect of different parameters on degree of hydration curve
(The units of S, K and D are h-2, µm/h, and µm2/h, respectively)
5.4.3 Effect of the number of initial nuclei
Garrault (2001) has found that the quantity of C-S-H nuclei precipitated on a C3S sample
varied with the lime concentration of the solution that the sample was suspended in. Specifically,
the total quantity (N) of precipitated C-S-H for 20 g of C3S was found to be 50 µmol in a solution
with a constant lime concentration of 11 mmol/L and 23 µmol in a saturated lime solution (22
mmol/L), resulting in a ratio of 2.17 for the initial number of nuclei. Samples pretreated for 30
minutes at these two different lime concentrations (presumably wearing different amounts of
initial nuclei) were further hydrated in saturated lime solutions to study the effect of initial nuclei
on hydration kinetics. Therefore, the proposed model was fitted to experimental data (Garrault
2001) by assuming only the number of initial C-S-H nuclei per unit surface area (which is
proportional to S) was different for the two samples. Since the particle size distribution data of
the C3S sample was not published, a characteristic particle size had to be assumed. It turned out
that there existed one set of best fit model parameters for each assumed representative particle
size. Experimental and fitted results (performed with two different representative particle sizes)
116
are presented in Figure 5.9, where experimental curves were offset 30 minutes to the right to take
into account the pretreatment time. As shown in the figure, nearly identical modeled curves can
be obtained by changing particle size and model parameters simultaneously. The values of the
model parameters obtained for different characteristic particle sizes are presented in Table 5.2.
The modeled ratio of n, which is equivalent to the ratio of S, for the two samples (assuming
constant g1) remains constant at 2, nearly identical to the experimental result of 2.17. These
results further confirm the validity of the model. The samples are most likely to have a
characteristic particle radius of 4 to 5 µm as the fitted model parameters for this size range are
closest to those obtained earlier for the same lime concentration.
0.25
Experimental Results
Modeled Results
Degree of Hydration
0.2
0.15
0.1
N=50µmol
N=23µmol
0.05
0
0
100
200
300
400
500
Time (min)
600
700
800
Figure 5.9: Experimental (Garrault 2001) and fitted hydration kinetics of C3S in saturated lime
solution with different quantities of C-S-H nuclei
117
Table 5.2: Model Parameters corresponding with different characteristic particle sizes
Assumed Characteristic
Particle Radius, R0 (µm)
S=ng12 (h-2)
K=g2/c (µm/h)
D (µm2/h)
3.5
0.0244
0.050
0.0015
4
4.5
5
0.0122 0.0224 0.0112 0.0202 0.0101 0.0180 0.0090
0.050 0.057 0.057 0.065 0.065 0.071 0.071
0.0015 0.0021 0.0021 0.0027 0.0027 0.0036 0.0036
5.4.4 Further discussion
Since the model developed in this study is based on a much simplified system of cement
hydration, i.e. C3S hydration in stirred dilute suspensions with lime concentration control, it is
natural to question its applicability to real systems such as Portland cement paste and C3S paste
hydration.
The first question that is typically raised is whether cement particles can be assumed to
hydrate individually without interacting with each other. Admittedly, in a paste, C3S particles are
likely to be in contact with each other rather than suspended in the solution. However, the
contacts are typically made by “points” with negligible area and this is probably why 2-D
microstructure images of C3S paste usually do not show connections between particles (Costoya
2008). In addition, for a w/c ratio of 0.4, each unit volume of C3S (whose approximate density is
3.15 g/cm3) has an average of 1.26 unit volumes of free space filled with water. For a spherical
particle with a radius of R0, this means an average distance of 0.31 R0 outward from its surface to
the next particle. The thickness of C-S-H formed on a C3S particle surface near the peak
hydration rate (when the transition from the NG stage to the DC stage starts to occur) is less than
0.4 µm (Costoya 2008), implying a maximum net growth of 0.15 µm after taking into account
dissolution. Considering that the particle sizes (diameter) of cement typically range from 1 to
100 µm, the growth is equivalent to 0.3 to 0.003 R0, depending on particle size. Therefore, C3S
particles are not likely to seriously interfere with each other during the NG stage of hydration.
118
Another question that should be considered is whether impingement between growing C-SH and CH should be taken into account. As discussed earlier, CH typically nucleates and grows
in the pore spaces while C-S-H nucleates and grows on particle surfaces. Therefore, the
interactions between CH and C-S-H are similar to interactions between different C3S particles
and can also be ignored during the NG stage of hydration. These are probably the reasons why
the model developed in this study can be successfully applied to Portland cement paste hydration,
as will be shown in Section 5.5. There is no doubt that different C3S particles as well as CH
crystals will eventually interfere seriously with each other, but most likely only during the DC
stage of hydration.
In some sense, a hydrating C3S particle is similar to a growing nucleus because the total
size of the particle gradually increases (hydration of 1 unit volume of C3S produces 1.569 unit
volumes of C-S-H). The only difference is that the growth rate is not constant due to the change
of the hydration mechanisms as well as the increasing thickness of the diffusion barrier. During
the DC stage, the space inside the diffusion barrier is apparently not enough to accommodate all
C-S-H hydration products. Therefore, while a portion of the dissolved ions is forming the inner
C-S-H, the remaining portion has to diffuse outward to the surface of the particle to form the
outer C-S-H. For this reason, it is still not clear whether the diffusion of water or the diffusion of
ions (such as H2SiO42-) is the rate controlling mechanism during the DC stage of hydration.
When C3S particles are considered as growing nuclei, the extended volume concept may be
applied for a second time to take into account the interactions among different C3S particles and
CH crystals. Since the total degree of hydration is proportional to the total volume of C-S-H, the
following equation may be derived,
119
d χ T = dα T ⋅
VT − VS
VT
(5.28)
where χT is the actual degree of hydration of the C3S paste; αT is the extended degree of
hydration of the C3S paste assuming no impingement among different particles and CH crystals
(i.e. the total degree of hydration as calculated by the current model); VT is the total
representative volume of a C3S paste while VS is the total volume of the solid phase in the
representative volume. However, whether the reduction factor to the total degree of hydration
can be evenly distributed to each individual particle to make Eq. (5.28) compatible with the
current model still deserves further investigation. In addition, Eq. (5.28) does not take into
account the shape of the CH crystals, which is known to be different in Portland cement paste
than in alite paste (Kjellsen 2004, Gallucci 2007, Costoya 2008).
5.5 Model Application: Class H Cement Paste Hydration
5.5.1 Model modifications and application procedures
Since the hydration of Portland cement follows a similar mechanism as that of C3S, the
model developed in this chapter can also be used to study cement hydration. As discussed in
Chapter 4, the early hydration of Portland cement is mainly attributed to C3S and C3A. Therefore,
the hydration kinetics curves (derivative curves) of cement that contains C3A typically exhibit
two peaks, which are very difficult to model accurately. Two out of the six different types of
cement (Class H-II and H-P) used in this study have zero C3A content and their experimental
results are used to fit the proposed model. The particle sizes of these cements were measured to a
very small scale (0.14 µm) compared with those of the C3S samples (2 µm) in Section 5.4. When
very fine particles are taken into account, it is likely that some particles may have completely
hydrated before the nucleation and growth stage begins and hence may not be applicable to the
120
model. The smallest 2% particles of the cements were not included when applying Eq. (5.27) to
experimental data, which roughly corresponds to particles smaller than 0.6 µm for Class H-II
cement and 0.5 µm for Class H-P cement. When modeling cement paste hydration, the most
important modification that needs to be made to the original model is introducing an offset time
and degree of hydration (although these values are sometimes found to be zero) to account for
the period before the nucleation and growth stage begins, during which the true hydration
mechanisms are still not fully understood. Fitting the model to experimental data consists of the
following four steps (Figure 5.10):
Step 1: Adjust the parallel growth rate constant S and the consumption rate constant K such that
the accelerating stage of the modeled derivative curve is parallel to that of the experimental
curve.
Step 2: Offset the starting time of the modeled curve from 0 to t0 such that the accelerating stage
of the modeled curve and that of the experimental curve almost coincide. Adjust the diffusion
constant D to fit the deceleration stage of hydration.
Step 3: Plot hydration rate against degree of hydration using both experimental and modeled
results and obtain the offset degree of hydration (α0).
Step 4: After applying α0, compare the modeled integral curve with the experimental one and
further adjust the diffusion constant D to achieve the best fit.
121
Experimental
0.025
0.02
Step 1
0.015
0.01
0.005
0
0
20
40
Time (h)
Rate of hydration (/h)
Rate of hydration (/h)
0.025
Modeled
0.02
Step 2
0.015
0.01
0.005
0
60
0
20
40
Time (h)
60
0.02
Step 3
0.015
0.01
0.005
0
0
0.1 0.2 0.3 0.4
Degree of hydration
Degree of hydration
Rate of hydration (/h)
0.025
0.5
0.4
Step 4
0.3
0.2
0.1
0
0
20
40
Time (h)
60
Figure 5.10: Example of fitting the proposed model to experimental data of Test H-II-5-1
5.5.2 Effect of curing temperature
Figure 5.11 shows the modeled hydration kinetic curves (derivative curves) compared with
those obtained experimentally for the Class H-II cement cured at different temperatures and a
constant pressure of 13.1 MPa. The integral curves of the Class H-II cement for all curing
conditions in Series I of pressure cell tests are compared with the fitted model in Figure 5.12.
Note that the experimental integral curves are the averaged test results and the experimental
derivative curves are obtained by polynomial fits as discussed in Section 2.4.1. The fitted model
parameters are presented in Table 5.3. It is found that all three rate constants (S, K, and D)
increase dramatically as the curing temperature increases. The offset time increases from 0 to 1.5
122
hours with increasing curing temperature while the offset degree of hydration decreases from
0.01 to 0 with increasing temperature.
60 °C
40.6 °C
Amb.
0.025
Rate of hydration (/h)
Experimental
Experimental
0.06
Modeled
0.02
0.015
Experimental
0.15
Modeled
0.04
0.1
0.02
0.05
Modeled
0.01
0.005
0
0
0
10
20
30
Time (h)
40
50
0
10
20
Time (h)
30
0
40
0
10
20
Time (h)
30
Figure 5.11: Experimental and modeled hydration kinetics of Class H-II cement
(w/c = 0.4, curing pressure = 13.1 MPa)
Experimental
13.1 MPa
6.9 MPa
Curing Pressure = 0.69 MPa
Degree of hydration
Modeled
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
60°C
0.3
40.6°C
0.2
60°C
0.3
0.2
0
Amb.
0.1
0
10
20
30
Time (h)
40
50
0
40.6°C
0.2
Amb.
Amb.
0.1
60°C
0.3
40.6°C
0.1
0
10
20
30
Time (h)
40
50
0
0
10
20
30
Time (h)
40
Figure 5.12: Experimental and modeled hydration kinetics of Class H-II cement cured at
different temperatures and pressures (w/c = 0.4)
50
123
Table 5.3: Dependence of model parameters on curing condition (Class H-II cement, w/c =0.4)
Temp. (°C)
Amb. (~24)
40.6
60
Pressure (MPa) 0.69
6.9
13.1
0.69
6.9
13.1
0.69
6.9
13.1
Test No.
24-I
24-II/III 24-IV
40-I
40-II/III 40-IV
40-I
40-II/III 40-IV
2
-2
S=ng1 (h )
0.0036 0.0034 0.0036 0.0220 0.0220 0.0240 0.0900 0.1100 0.1300
K=g2/c (µm/h) 0.060
0.059
0.062 0.150
0.150
0.160 0.420
0.460
0.430
2
D (µm /h)
0.03
0.03
0.03
0.10
0.10
0.10
0.23
0.24
0.23
t0 (h)
0
0
0
1
1
0.9
1.6
1.7
1.6
α0
0.01
0.01
0.01
0.006
0
0.01
0
0
0
As discussed in Section 4.3.1, the rate constant of cement hydration at an arbitrary curing
condition (T, P), can be related to that of a reference condition (Tr, Pr) by the following equation,
C (T , P ) =
k (T , P )
kr (Tr , Pr )
= C (T ) ⋅ C ( P )
(5.29)
where C(T) is a scale factor on the rate constant due to temperature change from the reference
temperature (Tr) to the arbitrary temperature (T) for a constant pressure process while C(P) is a
scale factor on the rate constant due to pressure change from the reference pressure (Pr) to the
arbitrary temperature (P). The scale factors are related to the curing conditions by:
k ′ (T , Pr )
C (T ) =
E
= exp  a
 R
kr (Tr , Pr )

 1 1 
 −  
 Tr T  
(5.30)
C ( P) =
k (T , P )
 ∆V ‡

= exp 
Pr − P ) 
(
k ′ (T , Pr )
 RT

(5.31)
For the model proposed here, the rate constants (nuclei growth rates g1 and g2) during the
NG stage of hydration cannot be directly derived by fitting the model with the experimental data
because the total number of nuclei per unit surface area (n) and the volume ratio (c) of the
hydration product to the reacted cement are still unknown. However, if both n and c are assumed
to be independent of curing conditions, then the scale factors on the growth rates due to different
124
curing conditions can be obtained from fitted values of S and K, according to their definitions.
On the other hand, the scale factor on the diffusion constant for the DC stage of hydration can be
obtained directly. Therefore,
C1 =
g1
S
=
g1r
Sr
(5.32)
C2 =
g2
K
=
g2r K r
(5.33)
D
Dr
(5.34)
C3 =
where C1 is the scale factor on the parallel growth rate (g1), C2 is the scale factor on the
perpendicular growth rate (g2), while C3 is the scale factor on the diffusion constant (D); the
subscript r represents the reference condition. Table 5.4 shows the scale factors obtained for
different curing conditions using the ambient curing temperature as the reference condition. The
scale factors are found to be similar for the same temperature change with some variations,
suggesting that curing temperature has slightly different effects on the different model
parameters. For example, the perpendicular growth rate appears to increase slightly faster than
the parallel growth rate with increasing curing temperature. However, the variations may also be
the result of inaccuracies of the estimated proportionality constants (Table 3.4) between chemical
shrinkage and degree of hydration (CS0), which strongly influence the converted experimental
data. In addition, it should be pointed out that the determination of the best fit parameters is
somewhat subjective. According to Eq. (5.30), the activation energy can be obtained by a linear
regression analysis using the scale factors and the temperature data. The results are shown in
Figure 5.13. All three scale factors, C1, C2 and C3, are found to satisfy Eq. (5.30) with activation
energies of 39.1, 45.3 and 46.7 kJ/mol, respectively. The dependence of activation energy on
125
curing pressure is too small to be effectively quantified with current test data. A better
temperature control scheme and a higher pressure range are required for such purpose.
Table 5.4: Scale factors on nuclei growth rate derived from fitted parameters
Temperature (°C)
Pressure
Scale factor
(MPa)
~241) 40.6
60
C1(T)
1
2.45 5.00
0.69
C2(T)
1
2.50 7.00
C3(T)
1
3.30 7.60
C1(T)
1
2.53 5.66
6.9
C2(T)
1
2.56 7.80
C3(T)
1
3.30 8.00
C1(T)
1
2.58 6.01
13.1
C2(T)
1
2.59 6.98
C3(T)
1
3.50 7.60
1)
: Assumed lab temperature
2.5
2.5
Data
Linear fit
2
2
0.5
1.5
ln(C3(T))
1
1
0.5
0
1
2
3
-1
1/Tr - 1/T (K )
4
-4
x 10
0
Data
Linear fit
2
1.5
ln(C2(T))
ln(C1(T))
1.5
0
2.5
Data
Linear fit
1
0.5
0
1
2
3
-1
1/Tr - 1/T (K )
4
-4
x 10
0
0
1
2
3
-1
1/Tr - 1/T (K )
4
-4
x 10
Figure 5.13: The temperature dependence of the scale factors (Class H-II cement, w/c = 0.4)
126
5.5.3 Effect of curing pressure
Figure 5.14 shows the modeled hydration kinetic curves compared with those obtained
experimentally for the Class H-II cement cured at different pressures and ambient temperature.
The results of Test H-II-3 are not included in the figure because they almost overlap with those
of Test H-II-2, probably due to the fact that the former was tested at a lower ambient temperature.
Unfortunately, the temperature of Test H-II-3 was not recorded. The fitted model parameters are
presented in Table 5.5. The effect of curing pressure on the rate constants (S and K) during the
NG stage of hydration is found to be similar to that of curing temperature, but at a much smaller
magnitude. On the other hand, in contrast to curing temperature, curing pressure is found to have
little, if any, effect on the diffusion constant (D) during the DC stage of hydration.
0.05
0.6
Experimental
Modeled
17.2 MPa
0.5
51.7 MPa
Degree of hydration
Rate of hydration (/h)
0.04
0.03
17.2 MPa
0.02
0.01
51.7 MPa
0.4
0.3
0.69 MPa
0.2
0.1
Experimental
Modeled
0.69 MPa
0
0
10
20
Time (h)
30
40
0
0
10
20
30
40
Time (h)
50
60
Figure 5.14: Experimental and modeled hydration kinetics of Class H-II cement cured at
different pressures and at ambient temperatures (w/c = 0.38)
70
127
Table 5.5: Dependence of model parameters and scale factors on curing pressure
(Class H-II cement, w/c =0.38)
Test No.
H-II-1
H-II-2
H-II-3 H-II-4
Curing Pressure (MPa)
0.69
17.2
34.5
51.2
Curing Temperature (°C)
26.1
26.1
26.7
S=ng12 (h-2)
0.0036 0.0049 0.0049 0.0105
K=g2/c (µm/h)
0.062
0.076
0.076
0.125
2
D (µm /h)
0.028
0.028
0.027
0.039
t0
0.5
0.7
0.2
0.3
α0
0.008
0.008
0.008
0.01
C1(T, P)
1
1.23
1.23
1.80
C2(T, P)
1
1.23
1.23
2.00
C1(T)
1
1
1.031)
C2(T)
1
1
1.042)
C1(P)
1
1.23
1.74
C2(P)
1
1.23
1.93
1)
: Estimated according to the obtained activation energy of 39.1 kJ/mol.
2)
: Estimated according to the obtained activation energy of 45.3 kJ/mol.
Table 5.5 also shows the obtained scale factors (due to the combined effect of temperature
and pressure) calculated from the fitted parameters according to Eqs. (5.32) and (5.33) for
different curing conditions using (26.1°C, 0.69MPa) as the reference one. Note that C3 is not
calculated here because the diffusion constant does not appear to depend on curing pressure. To
account for lab temperature fluctuations, the effect of temperature on the scale factor C(T), can
be estimated from the previously derived activation energies and separated from that of the
pressure (C(P)). The results are also given in Table 5.5. The perpendicular growth rate also
appears to increase slightly faster than the parallel growth rate with increasing curing pressure.
As shown in Figure 5.15, both C1 and C2 are found to satisfy Eq. (5.31), with activation volumes
of -26.8 and -30.4 cm3/mol, respectively.
128
0.8
0.8
Data
Linear fit
Data
Linear fit
0.6
ln(C2(P))
ln(C1(P))
0.6
0.4
0.2
0
-60
0.4
0.2
-50
-40
-30
-20
Pr - P (MPa)
-10
0
0
-60
-50
-40
-30
-20
Pr - P (MPa)
-10
0
Figure 5.15: The pressure dependence of the scale factors (Class H-II cement, w/c = 0.38)
5.5.4 Effect of w/c ratio
Figure 5.16 shows the experimental and modeled hydration kinetics of the Class H-II
cement prepared with two different w/c ratios and cured at three different pressures. The fitted
model parameters as well as the derived scale factors are shown in Table 5.6. Since the ambient
temperatures of these tests were not recorded, it is impossible to separate the effect of curing
pressure from that of curing temperature. The derivative curves of samples made with different
w/c ratios are found to have similar shapes. The slight variations in the peak rate are probably
due to lab temperature fluctuations. Correspondingly, the fitted rate constants (S and K) for the
nucleation and growth controlled stage are found to be independent of w/c ratio with some small
random variations. The kinetics curves of the slurries with the lower w/c ratio (0.3) are offset to
the left compared with those with the higher w/c ratio (0.5), suggesting that the former entered
the acceleration stage (NG stage) of hydration at an earlier time than the latter. The difference in
the modeled offset time (t0) for the two w/c ratios ranges from 1 to 2.9 hours and decreases with
increasing curing pressure. Due to these offsets, the total degree of hydration of cement is found
129
to increase with decreasing w/c ratio at early ages (t < 20 h). Similar results were observed in
other studies (Zhang 2010, Baroghel-Bouny 2006), though the differences might be very small
and difficult to identify sometimes (Zhang 2010). Derivative curves obtained with the isothermal
calorimetry method generally show little or no such offset (Thomas 2011, Sandberg 2005, Bentz
2009). Considering the small sample size (< 5 g) typically used in isothermal calorimetry tests,
the test results given here (for sample size > 2 kg) are probably closer to the conditions in real
applications. At latter ages, the total degree of hydration is found to decrease with decreasing w/c
ratio, consistent with previous studies of both chemical shrinkage tests and isothermal
calorimetry tests (Zhang 2010, Bentz 2009). Correspondingly, the fitted diffusion constant (D)
decreases with decreasing w/c ratio, consistent with another study that applies the diffusion
model to C3S hydration using a single representative particle size (Berliner 1998).
In cement pastes with lower w/c ratios, cement particles are more densely packed and have
smaller inter-particle distances. The values of the fitted parameters suggest that, for the range
studied here, interactions between cement particles are minimal during the NG stage of hydration
and that the nuclei growth rates are not affected by the inter-particle distance. However, as
hydration progresses, the limited inter-particle space leads to larger total contact area between
cement particles (with hydration products on the surface), which effectively reduces the contact
area between capillary water and the particles and hence reduces the apparent diffusion rate.
Therefore, the fitted value of the diffusion constant is lower for pastes with lower w/c ratios.
Since the contact area between cement particles also increases with increasing degree of
hydration, adopting a constant diffusion constant to model the DC stage of hydration is only an
approximation and may result in discrepancies when long term hydration is considered.
130
Rate of hydration (/h)
Curing pressure = 0.69 MPa
Curing pressure = 17.2 MPa
Experimental
Modeled
0.03
Experimental
Modeled
0.03
0.02
0.02
0.01
0.02
0.01
w/c=0.3
w/c=0.3
10
Degree of hydration
20
30
0
40
w/c=0.5
0.5
0
w/c=0.3
0.2
Experimental
Modeled
0
20
40
Time (h)
30
40
20
30
40
w/c=0.5
0.3
0.2
w/c=0.5
Experimental
Modeled
0.1
0
10
w/c=0.3
0.3
60
0
0.4
w/c=0.3
0.2
0.1
20
0
0.5
0.4
0.3
10
w/c=0.3
0.5
0.4
0
w/c=0.5
0.01
0
Experimental
Modeled
0.03
w/c=0.5
w/c=0.5
0
Curing pressure = 34.5 MPa
0
20
40
Time (h)
60
Experimental
Modeled
0.1
0
0
20
40
Time (h)
60
Figure 5.16: Experimental and modeled hydration kinetics of Class H-II cement with different
w/c ratios and cured at different pressures
Table 5.6: Dependence of model parameters and scale factors on w/c ratio (Class H-II cement)
Curing
0.69
17.2
34.5
Pressure (MPa)
w/c ratio
0.3
0.5
0.3
0.5
0.3
0.5
2
-2
S=ng1 (h )
0.0027 0.0031 0.0046 0.0041 0.0061 0.0068
K=g2/c (µm/h) 0.055 0.057 0.072 0.066 0.083 0.085
D (µm2/h)
0.016 0.036 0.027 0.036 0.017 0.036
t0
-0.8
2.1
-0.7
1.1
-0.2
0.8
α0
0.006
0.02
0.008
0.01
0.01
0.01
C1(T, P)
1
1
1.3
1.16
1.50
1.49
C2(T, P)
1
1
1.3
1.16
1.50
1.49
5.5.5 Effect of cement composition
Figure 5.17 shows the experimental and modeled hydration kinetics (integral curves) of the
Class H-P cement cured at different temperatures and pressures. While the model appears to give
131
perfect fits to experimental data obtained at ambient temperatures (and different pressures), it
seems to overestimate the degree of hydration at higher curing temperatures during later ages.
There are two possible reasons for the discrepancies. The first one is that the Class H-P cement
contains a significant amount of C2S while the model was formulated based on C3S hydration.
The second one is that the apparent diffusion constant probably decreases with increasing degree
of hydration as a result of inter-particle interactions (which become more significant at higher
degrees of hydration) while the model adopted a constant diffusion constant.
The fitted model parameters and the calculated scale factors are listed in Table 5.7.
Negative values of offset time are obtained for the Class H-P cement at ambient temperatures,
compared with positive values for the Class H-II cement (Table 5.5), indicating that the former
entered the nucleation and growth stage at an earlier time than the latter. The dependence of the
scale factors on curing temperature and pressure are shown in Figure 5.18. The activation
energies determined from C1, C2 and C3, are 39.4, 46.7 and 34.4 kJ/mol, respectively. The
activation volumes determined from C1 and C2 are -28.3 and -29.2 cm3/mol, respectively. The
values are remarkable close to those determined for Class H-II cement in Sections 5.5.3 and
5.5.4, except for the activation energy of the diffusion constant determined from C3, which is
about 25% lower than that of the Class H-II cement. Similar to Class H-II cement, the
dependence of diffusion constant (D) on curing pressure seems to be too small to be effectively
quantified.
132
Curing temperature 22-26 °C
Curing pressure 0.69 MPa
0.6
0.7
Degree of hydration
0.5
0.6
0.4
0.5
51.7 MPa
0.3
17.2 MPa
0.2
0.1
0
0.4
34.5 MPa
Experimental
Modeled
0
10
20
30
40
Time (h)
50
60 °C
0.3
40.6 °C
0.2
25.6 °C
Experimental
Modeled
0.1
60
70
0
0
10
20
30
40
Time (h)
50
60
Figure 5.17: Experimental and modeled hydration kinetics of Class H-P cement cured at
temperatures and pressures (w/c = 0.38)
Table 5.7: Model parameters and scale factors for Class H-P cement (w/c = 0.38)
Test No.
H-P-1 H-P-2 H-P-3 H-P-4 H-P-5 H-P-6
Curing Pressure (MPa)
0.69
17.2
34.5
51.2
0.69
0.69
Curing Temperature (°C) 25.6
22.2
23.9
26.1
40.6
60
2
-2
S=ng1 (h )
0.0025 0.0028 0.0053 0.0087 0.0170 0.0680
K=g2/c (µm/h)
0.069 0.069 0.092 0.130 0.186 0.483
D (µm2/h)
0.072 0.060 0.072 0.099 0.132 0.300
t0
-1.1
-1.6
-0.8
-0.4
0.7
1.4
α0
0.004
0
0
0
0.005 0.006
1)
1)
1)
C1(T)
1
0.83
0.91
1.03
2.60
5.20
2)
2)
2)
C2(T)
1
0.81
0.90
1.03
2.70
7.00
C3(T)
1
1.83
4.17
C1(P)
1
1.27
1.59
1.81
1
1
C2(P)
1
1.25
1.49
1.80
1
1
1)
: Estimated according to the obtained activation energy of 39.4 kJ/mol.
2)
: Estimated according to the obtained activation energy of 46.7 kJ/mol.
70
133
2.5
0.8
2
ln(C1(T))
0.7
ln(C2(T))
0.6
ln(C1(P))
ln(C2(P))
Linear fits
1.5
0.5
Linear fits
ln(C(P))
ln(C(T))
ln(C3(T))
1
0.4
0.3
0.2
0.5
0.1
0
0
1
2
3
1/Tr - 1/T (K-1)
0
-60
4
-4
x 10
-40
-20
Pr - P (MPa)
0
Figure 5.18: The temperature and pressure dependence of the scale factors
(Class H-P cement, w/c = 0.38)
Since the rate constants (S, K and D) are strongly affected by the curing temperature and
pressure, it is useful to derive the average values for a fixed curing condition (reference condition)
to study their dependence on cement composition. The parameters for a fixed curing condition
can be obtained by rewriting Eqs. (5.32) - (5.34) as:
Sr =
S
C12
Kr =
K
C2
(5.36)
Dr =
D
C3
(5.37)
(5.35)
C1 and C2 can be estimated according to Eq. (5.29) while C3 can be estimated according to Eq.
(5.30) (assuming D is independent of curing pressure) using the previously derived activation
energies and activation volumes. A temperature of 24 °C was assumed for tests performed at
uncertain ambient temperatures. The average value of Sr (24 °C, 0.1 MPa) is found to be
134
0.0030±0.0004 h-2 for Class H-II cement and 0.0024±0.0004 h-2 for Class H-P cement. The
average value of Kr (24 °C, 0.1 MPa) is found to be 0.054±0.003 µm/h for Class H-II cement and
0.063±0.002 µm/h for Class H-P cement. Note that Sr and Kr are independent of w/c ratio. For
w/c = 0.38, the average value of Dr (24 °C, 0.1 MPa) is found to be 0.028±0.004 µm2/h for Class
H-II cement and 0.067±0.012 µm2/h for Class H-P cement.
In summary, the Class H-P cement, which has a much lower C3S content and a much
higher C2S content than the Class H cement, is found to have a slightly lower (20%) value of S, a
slightly higher (17%) value of K, and a significantly higher (139%) value of D. Considering that
S = ng12, decreasing C3S content appears to decrease either the number of nuclei per unit surface
area (n) or the parallel growth rate of the nuclei (g1). Assuming a density of 3.15 g/cm3 for C3S,
3.28 g/cm3 for C2S, and 2.0 g/cm3 for C1.7SH4 (Bishnoi 2009a), then c equals to 1.569 and 2.166
for C3S and C2S, respectively. Considering K = g2/c, decreasing C3S content appears to increase
the perpendicular growth rate of the nuclei. Since the diffusion constant is directly related to the
porosity of the diffusion barrier, decreasing C3S content probably increases the porosity of the
inner layer of hydration products on the surface of the anhydrous core. When pure C3S is
hydrated in stirred dilute suspensions with lime concentration control to prevent nucleation of
CH (Section 5.4), the diffusion constant is about one order of magnitude lower than those
obtained for Class H-II and H-P cement pastes hydration.
5.6 Implications for Cement Hydration Mechanisms
The two traditional NG-based models (Brown 1985, Thomas 2007) currently used for
cement hydration were both originally developed for solid phase transformations. According to
these models, hydration rate is directly associated with the size of the growth front of the nuclei,
135
which initially increase as a result of increasing size (and number) of nuclei and subsequently
decrease due to interferences between different nuclei hence generating a bell-shaped rate curve
similar to that of cement hydration. However, with these models, the w/c ratio would be expected
to significantly change the hydration rate during the NG stage since the available space to
accommodate growth is greatly affected. However, as discussed earlier, test results from both
chemical shrinkage and isothermal calorimetry showed that the shape of the hydration kinetics
curves during early periods are largely independent of w/c ratio. To eliminate the model’s
dependence on w/c ratio, Thomas (2007) restricted the total volume available for growth to the
“hydration volume,” defined as the volume occupied by the hydration products (excluding
porosity) after complete hydration, without further justification for such treatment. In addition,
both of the traditional models require the initial volume available to be fictitiously reduced using
some scaling parameters, which is not substantially justified from an experimental point of view
(Xie 2011).
The fitted results in this section show that cement paste hydration can be simulated with the
same model developed based on C3S hydration in dilute suspensions by simply introducing two
parameters to offset the starting point of the modeled curve and account for the hydration at very
early periods (before the end of the induction period) whose mechanism is still not well
understood. Therefore, the study not only confirms the traditional belief that the hydration of
cement follows the same mechanism as that of its main component (C3S), but also suggests that
hydration of cement paste may be governed by a similar mechanism as in stirred dilute
suspensions where each particle reacts individually with no interferences between each other.
Nuclei interactions are only limited to the surfaces of the same cement particles. Therefore, the
w/c ratio that affects inter-particle spacing has little effect on hydration rate during the
136
acceleration stage (NG stage) of hydration, consistent with experimental observations. Interparticle interactions become significant only during later stages (DC stage) of hydration, which
may be indirectly accounted for by their effects on the apparent diffusion constant. As shown in
Figure 5.19, the modeled hydration rate of each individual particle does not decrease until it
enters the DC stage and the deceleration of the modeled total hydration rate (which
approximately corresponds to period 4 of the five periods of hydration) is a result of particles of
different sizes entering the DC stage at different times. While a good fit with experimental data
does not necessarily guarantee the assumptions made in the model are correct, the study
presented here certainly provides an alternative explanation of cement hydration mechanism.
Hydration kinetics of individual particles
Smallest particles entering DC stage
R0=0.5
0.14
R0=1
0.12
R0=2
0.1
R0=4
0.02
Rate of hydration (/h)
Rate of hydration (/h)
Total hydration kinetics
0.025
0.16
R0=8
0.08
R0=16
0.06
R0=64
0.04
0.015
0.01
0.005
All particles entered DC stage
0.02
0
0
5
10
15
Time (h)
20
25
30
0
0
5
10
15
Time (h)
20
25
30
Figure 5.19: Modeled hydration kinetics of individual cement particles and the weighted average
result of a sample with multiple particle sizes
(R0 is the particle radius in µm, ◊ indicate transition from NG to DC stage)
5.7 Summary
A particle-based numerical model that successfully combines the nucleation and growth
mechanism and the diffusion mechanism of cement hydration is developed in this chapter. The
137
model assumes that during the nucleation and growth stage of hydration C-S-H nuclei are
instantaneously precipitated on the spherical surfaces of cement particles on first contact with
water and that these nuclei grow parallel and normal to the cement particle surface at two
constant but different rates. Dissolution of the original cement particles and impingement
between different C-S-H nuclei on the same particle surfaces as hydration progresses are both
taken into account. It is further assumed in the model that hydration of a particular cement
particle enters the diffusion controlled stage as soon as the anhydrous core is completely covered
by hydration product. While such transition of the rate controlling mechanism of a single particle
may result in a steep reduction in its hydration rate, the deceleration of the overall hydration rate
of a cement sample is more gradual due to the fact that these transitions occur at different times
for particles of different sizes. The time at which a particle enters the DC stage is controlled by
its size and the number of nuclei on its surface. These results support an original school of
thought that diffusion control is the cause of the main hydration peak, which is rivals an
emerging school of thought that nucleation and growth control the entire early hydration period
encompassed by the main rate peak (Thomas 2007, Thomas 2011). These two rivaling
viewpoints shall be further investigated in the future.
The model requires the cement particle size distribution as input and produces hydration
kinetics as output. For the NG stage, hydration kinetics is controlled by four parameters, namely
the total number of nuclei per unit surface area (n), the parallel growth rate of the nuclei (g1), the
perpendicular growth rate of the nuclei (g2), and the volume ratio of the hydration products
(formed on the surface of the cement particles, mainly C-S-H) to the cement reacted (c). The first
two parameters are coupled as a parallel growth rate constant (S = ng12) while the last two
parameters are coupled as a consumption rate constant (K = g2/c) such that they can be
138
independent determined by fitting the model to experimental data. For the DC stage, hydration
kinetics is controlled by a single diffusion constant D. Therefore, the model involves a total of
two rate constants for the NG stage and one constant for the DC stage of hydration.
The model has been shown to fit experimental data of C3S hydration in stirred dilute
suspensions exceptionally well. The fitted value of S appears to increase with increasing particle
size, suggesting that the total number of nuclei per unit surface area probably increases with
increasing particle size. The fitted values of K and D appear to be independent of particle size.
Together with other experimental data, the fitted model parameters for C3S hydration in stirred
saturated lime solutions suggest that the total number of nuclei per unit surface area ranges from
0.5 to 0.8 µm-2, while the perpendicular growth rate of the nuclei ranges from to 0.08 to
0.12µm/h. The diffusion constant for the controlling diffusion mechanism is estimated to be
approximately 1.8 x 10-15 m2/h, independent of C3S particle size.
With some slight modifications, the model has also been shown to fit nearly perfectly with
the hydration kinetics data of certain Portland cement pastes (such as those prepared with Class
H oil well cements with no C3A contents). The fitted values of S and K vary significantly with
curing condition but appear to be independent of w/c ratio. Both K and the square root of S
follow basic chemical kinetics laws in terms of their dependencies on curing temperature and
pressure with slightly different activation energies and activation volumes, suggesting that n is
probably independent of curing condition while g1 and g2 follow the chemical kinetics laws. The
diffusion constant (D), has a similar dependence on curing temperature as the growth rates (g1
and g2); but its dependence on curing pressure appears to be too small to be effective quantified
with the current test data. The fitted value of D decreases with decreasing w/c ratio, suggesting
that the apparent diffusion rate is affected by inter-particle distances, which control the contact
139
area between different particles. Therefore, it is likely that D should be modeled as a function of
the degree of hydration and the porosity of the cement paste when long-term hydration is
considered. Adopting a constant value of D in the model is adequate to provide nearly perfect fits
to most experimental data of this study except during the later periods (t > 40 h) of Class H-P
cement hydration at high curing temperatures (T > 40 °C). The deviations may be attributed to
the high C2S content of the particular cement or the decreasing apparent diffusion rate with
increasing degree of hydration. Increasing C3S content of the cement appears to generate a less
porous inner layer of hydration products (diffusion barrier), which may result in slower diffusion
rate during the DC stage of hydration.
140
CHAPTER 6 : CORELATION BETWEEN CHEMICAL
SHRINKAGE AND HEAT OF HYDRATION OF CEMENT
6.1 Introduction
Since the automated chemical shrinkage test adopted in this study is a relatively new
method of evaluating cement hydration kinetics, it is important to investigate its correlation with
the traditional method, i.e. the isothermal calorimetry method, to assess its reliability. Chemical
shrinkage tests performed at 0.69 MPa are used to compare with the isothermal calorimetry tests
performed at atmospheric pressure (0.101 MPa). Note that the isothermal calorimetry data
presented in this Chapter were obtained by Dr. Dale Bentz. As discussed in Chapter 2, the
recorded temperatures of the chemical shrinkage tests in this study are not very accurate due to
the limitations of the temperature control scheme. Isothermal calorimetry data, which were
obtained at precise temperatures, can be used in this Chapter to further calibrate the model
developed in Chapter 4. Unfortunately, due to the fact that standard isothermal calorimeters do
not allow the application of hydrostatic pressures, only the effect of curing temperatures on
hydration can be investigated in this chapter.
Among the different methods of evaluating cement hydration kinetics, isothermal
calorimetry used to be the only one that gives continuous test results. Therefore, correlations
between different methods are traditionally evaluated at discrete data points (Parrott 1990, Bentz
1995, Escalante-Garcia 2003, Zhang 2010). Costoya (2008) compared the continuous hydration
kinetics curves of alite measured by chemical shrinkage and by isothermal calorimetry for a
period of about 20 hours, and found nearly perfect agreement in test results. However, the
correlation between test results of these two different methods is more complicated for Portland
141
cement due to its multiphase characteristics. As discussed in Section 1.2.2, estimating the
normalization factors in Eq. (1.2), i.e. wn0, H0, and CS0, is essential for studying the correlations
between different indirect methods to evaluate cement hydration kinetics. For a given cement,
wn0 depends on the molecular weights of the hydration products and H0 depends on the
enthalpies of the hydration reactions, both of which are expected to remain constant as long as
the chemical formulae of the hydration products do not change. CS0 depends on the molecular
volumes of capillary water and the hydration products, and hence varies with both temperature
and pressure. While the ratio of wn0 to CS0 may be assumed to be independent of cement
composition due to the fact that specific volumes of non-evaporable water in different hydration
products are more or less the same (Chapter 3), the ratio of H0 to CS0 varies with cement
composition due to the fact that the enthalpies of different reactions are not related to the volume
changes caused by these reactions. H0/CS0 has been found to range from 65 to 85 (J/g
cement)/(mL water per 100 g cement), or 6500 to 8500 J/mL, at ambient temperatures for
different cements using the traditional discontinuous chemical shrinkage test method (Bentz
2008, Bentz 2010).
Similar to Eqs. (3.1) and (3.2), the cumulative heat evolution of cement at complete
hydration can be modeled by the following equation,
H
0
= c1 ⋅ p C 3 S + c 2 ⋅ p C 2 S + c 3 ⋅ p C 3 A + c 4 ⋅ p C 4 A F
(6.1)
where ci is a constant that is equal to the total amount of heat liberated associated with the
complete hydration of 1 g of the i-th compound in Portland cement while pi is the Bogue weight
fraction as defined in Section 1.2.2. These coefficients (ci) can be estimated theoretically by
calculating the enthalpies of the hydration reactions using the standard enthalpies of formation of
the relevant compounds involved. Alternatively, they can also be estimated by performing a
142
multi-linear regression analysis using H0 data obtained for a number of different cements
according to Eq. (6.1). Note that although the rate of heat evolution during cement hydration at
later ages is too low to be detected by conduction calorimeter, the total amount of heat liberated
at any time can be determined by the heat of solution in acid (Lerch 1948). Table 6.1 shows that
the coefficients for H0 obtained from these two different methods agree well with each other.
Table 6.1: Coefficients for total heat of hydration H0, in J/g (Taylor 1997a)
Compound
C3S
C2S
C3A
C4AF
a
Experimental
510
247
1356
427
Enthalpy of Complete Hydration 517 ± 13
262
1144b-1672c
418d
a
: Obtained by multi-linear regression analysis from experimental data (w/c=0.4, age
= 13years, cured at 21°C)
b
: Reaction with gypsum to give monosulfoaluminate
c
: Reaction with gypsum to give ettringite
d
: Reaction in the presence of excess calcium hydroxide to give hydrogarnets
6.2 Preliminary Analysis of Test Data
The one-parameter model developed in Chapter 4 is based on the assumption that for a
given cement paste the differential equation curves (rate of hydration vs. degree of hydration)
obtained at different curing conditions converge to a universal curve when normalized. For the
effect of curing pressures, this assumption has been verified for all the different type of cement
used in this study. However, for the effect of curing temperatures, it was verified only for Class
H-II cement due to difficulties of calculating reliable rates of hydration from chemical shrinkage
data at high curing temperatures. The rate of heat evolution can be directly measured by
isothermal calorimetry at different temperatures with very good accuracy. These heat evolution
data for different types of cement are converted to hydration kinetics and used here to further
check the validity of the assumption. The values of H0 can be estimated by the following
equation for the conversion of test data (Table 6.1),
143
H 0 = 510 p C 3 S + 247 p C 2 S + 1356 p C 3 A + 427 p C 4 AF + 239 p C 2 F
(6.2)
where the coefficient for C2F is obtained by assuming it generates the same amount of heat as
C4AF on the same mass basis. By substituting the Bogue compound fractions listed in Table 2.2,
the results calculated for Class A, C, G, H-I, and H-P cement are 497.7, 461.1, 470.1, 429.2, and
385.9 J/g cement, respectively.
Figure 6.1 shows the differential equation curves derived from the isothermal calorimetry
data of Class H-I cement before and after normalization. The rates of hydration obtained from
isothermal calorimetry data are much more accurate than those calculated from chemical
shrinkage data and do not oscillate. While the normalized differential equation curves at different
temperatures coincide relatively well during early and late periods, some deviations are observed
during the middle period, i.e. for degrees of hydration approximately ranging from 0.2 to 0.5. It
should be noted that the samples used for isothermal calorimetry studies reached higher final
degrees of hydration due to the longer curing period (7 days compared with 3 days for chemical
shrinkage tests). Figure 6.2 shows the normalized differential equation curves of four other types
of cement obtained at different curing temperatures. Similar convergence behaviors were
observed except for the test results of Class A and C cements at 60 °C, which diverge relatively
significantly from those obtained at 25 and 40 °C. These differences are mainly due to the fact
that the different compounds in cement hydrate at different rates and have slightly different
temperature sensitivities (activation energies). As the temperature difference increases, its
different effects on different compounds become more and more significant. The nearly perfect
convergences of test results at different curing pressures found for the different types of cement
in Chapter 4 are due to either the relatively small change in hydration rate or the relatively
uniform pressure sensitivities (activation volumes) of different compounds in cement.
144
0.14
1
25°C
25°C
40°C
Normalized rate of hydration
Rate of hydration (/h)
0.12
60°C
0.1
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
Degree of hydration
40°C
0.8
60°C
0.6
0.4
0.2
0
0.8
0
0.2
0.4
0.6
Degree of hydration
0.8
Figure 6.1: Effect of curing temperature on hydration rate as a function of degree of hydration
(Class H-I cement, w/c = 0.38)
Normalized rate of hydration
1
1
25°C
0.8
25°C
0.8
40°C
60°C
60°C
0.6
0.6
0.4
0.4
0.2
0
0.2
Class A
0
0
0.2
0.4
0.6
0.8
1
Normalized rate of hydration
40°C
Class C
0
0.2
0.4
0.6
1
25°C
0.8
25°C
0.8
40°C
40°C
60°C
60°C
0.6
0.6
0.4
0.4
0.2
0
0.8
0.2
Class G
0
0.2
0.4
0.6
Degree of hydration
0.8
0
Class H-P
0
0.2
0.4
0.6
Degree of hydration
0.8
Figure 6.2: Normalized differential equation curves of different types of cement
145
The results in Figure 6.2 also suggest that using the peak hydration rates to calculate the
scale factor C (as proposed in Chapter 4) is not always reliable because the peaks associated with
different phases may be superimposed on each other at high curing temperatures. According to
the model developed in Chapter 4, the hydration kinetics curve at curing temperature T2 can be
predicted by the one at T1 by the following equation,
α (t ) T = α (CT −T ⋅ t − t0 )
2
1
2
T1
(6.3)
where t0 is the offset time and C is the scale factor that can be expressed as,
E
CT1 −T2 = exp  a
 R
 1 1 
 −  
 T1 T2  
(6.4)
Therefore, the scale factor C can be estimated by transforming the hydration kinetics curve at T1
to achieve the best agreement with that at T2. Figure 6.3 shows the results of transforming the
experimental hydration kinetics curves obtained at 25 and 40 °C to predict those at higher curing
temperatures for Class C cement. Although it seems impossible to achieve perfect agreement for
the entire curing period, the predicted hydration kinetics curves are very accurate during the
period up to the first peak, which is mainly associated with C3S hydration. As discussed in
Chapter 4, the second peak is mainly due to C3A hydration, which has a higher value of
activation energy than C3S. Therefore, the transformed hydration kinetics curves typically
underestimate the second peak. As will be further explained in the next section, the problem may
be avoided if different scale factors were applied to different compounds in cement. As the
differences between the scale factors of different compounds increase with increasing
temperature difference, the predicted hydration kinetics curves also become less and less
accurate with increasing temperature difference.
146
1
0.25
Measured
Transformed from 25°C
0.2
0.6
60°C
40°C
Rate of hydration (/h)
Degree of hydration
0.8
25°C
0.4
Measured
0.2
Transformed from 40°C
0.15
60°C
0.1
40°C
25°C
0.05
Transformed from 25°C
Transformed from 40°C
0
0
20
40
Time (h)
60
80
0
0
5
10
15
Time (h)
Figure 6.3: Measured and predicted hydration kinetics at different curing temperatures by
coordinate transformations (Class C cement, w/c = 0.56)
Figure 6.4 shows the measured and predicted hydration kinetics curves for other types of
cement used in this study. It appears that the predictions are least accurate for Class A cement
possibly due to its high C3A content. On the other hand, the predictions for Class H-P and H-I
cements, both of which have little to no C3A content, are much more accurate. It is also
interesting to note that the transformed hydration kinetics curves from low temperatures typically
slightly overestimate the degree of hydration for high temperature tests at later ages, with the
exception of Class H-P cement, for which the opposite is true. The offset times, scale factors, as
well as the activation energies calculated from each scale factor using Eq. (6.12) for different
types of cement are listed in Table 6.2. The constants are obtained by trial and error to achieve
the best agreement as shown in Figures 6.3 and 6.4. Since it is impossible to achieve perfect
agreements, estimation of these values is somewhat subjective. For comparison purpose, the
scale factors and activation energies calculated from peak hydration rates as proposed in Chapter
4 are also listed in the Table. The values of C and Ea obtained by the two different methods
147
generally agree well with each other for relatively small temperature changes (from 25 to 40 °C
and from 40 to 60 °C) but may differ quite noticeably for the large temperature change (from 25
to 60 °C). The activation energies obtained in different temperature ranges appear to decrease
with increasing temperature except for Class A cement, for which the opposite is true. The
activation energies are also found to be much lower than those calculated in Chapter 4 from
chemical shrinkage test data. As mentioned previously, the later are probably inaccurate due to
the inadequate temperature control and errors associated with estimating CS0 at different
temperatures (H0 is independent of temperature).
Transformed from 25°C
0.7
0.7
0.6
0.6
0.5
60°C
0.4
40°C
0.3
25°C
Degree of hydration
Degree of hydration
Measured
0.2
Class A
0.1
0
0
50
Transformed from 40°C
0.5
60°C
0.4
40°C
25°C
0.3
0.2
Class G
0.1
100
0
150
0
50
Time (h)
150
0.8
Degree of hydration
Degree of hydration
0.8
0.6
60°C
0.4
40°C
25°C
0.2
0.6
60°C
0.4
40°C
25°C
0.2
Class H-P
0
100
Time (h)
0
50
Class H-I
100
Time (h)
150
0
0
50
100
150
Time (h)
Figure 6.4: Measured and predicted hydration kinetics of different cements at different curing
temperatures by coordinate transformations
148
Table 6.2: Activation energies obtained from different methods
T1 - T2
Cement
(°C)
Best fit method
(Eq. (6.3))
CT1 −T2
t0 (h)
Peak hydration rate method
(Eq. (4.16)a)
CT1 −T2
Ea (kJ/mol)
38.4
40.2
46.2
40.3
37.0
34.2
43.1
40.2
34.2
39.6
35.5
32.2
42.2
37.0
32.2
25-40
0.4
2.1
A
25-60
0.8
5.5
40-60
0.8
2.9
25-40 0.75
2.18
C
25-60
1
4.8
40-60
0.6
2.2
25-40
0.9
2.3
G
25-60
1
5.5
40-60
0.5
2.2
25-40
1
2.15
H-P
25-60
1.5
4.5
40-60
1.1
2.1
25-40
1
2.26
H-I
25-60
1.2
4.8
40-60
0.7
2.1
a
: T1 as the reference temperature
2.28
7.04
3.09
2.23
5.48
2.46
2.37
5.76
2.43
2.12
4.53
2.14
2.26
5.54
2.45
Ea (kJ/mol)
42.7
46.1
48.9
41.5
40.1
39.0
44.7
41.3
38.5
38.9
35.6
33.0
42.2
40.4
38.9
Ea (kJ/mol)
(from Ch.4)
52.6
48.8
50
42.5
44.3
6.3 Theoretical Analysis
In order to predict the effect of curing condition on cement hydration kinetics more
precisely, the model developed in Chapter 4 shall be applied to each individual clinker phases. If
the following equation is used to represent hydration kinetics (integral curve) for the reference
condition (Tr, Pr),
α ( t ) = pC Sα C S ( t ) + pC SαC S ( t ) + pC Aα C A ( t ) + pC AF α C AF ( t )
3
3
2
2
3
3
4
4
(6.5)
then the hydration kinetics under any other curing condition (T, P) can be represented by,
α ( t ) = pC SαC S ( CI t ) + pC SαC S ( CII t ) + pC AαC A ( CIII t ) + pC AFαC AF ( CIV t )
3
3
2
2
3
3
4
4
(6.6)
where Ci is a scale factor for each individual clinker phase, associated with the activation energy
(Ei) and activation volume (∆Vi‡) of the particular phase by,
149
E
Ci = exp  i
R
 1 1  ∆Vi ‡  Pr P  
 − +
 −  
T
T
R
 T T 
 r

(6.7)
Apparently, application of the model proposed above requires experimental hydration kinetics
curves for each individual clinker phase, which is not yet possible to measure continuously
during the hydration of Portland cement.
Similar to Eq. (6.5), experimental results of the indirect methods (chemical shrinkage and
heat evolution) of measuring cement hydration kinetics are more accurately represented by the
degree of hydration of each individual clinker phase as follows,
CS ( t ) = a1 pC3 S α C3 S ( t ) + a2 pC2 S α C2 S ( t ) + a3 pC3 Aα C3 A ( t ) + a4 pC4 AF α C4 AF ( t )
(6.8)
H ( t ) = c1 pC3S α C3S ( t ) + c2 pC2 S α C2 S ( t ) + c3 pC3 Aα C3 A ( t ) + c4 pC4 AF α C4 AF ( t )
(6.9)
Eqs. (6.8) and (6.9) reduce to Eqs. (3.1) and (6.1), respectively, at the complete hydration
condition (i.e. when the degrees of hydration of all four different phases are equal to 1). Since the
coefficients associated with different phases (ai for chemical shrinkage and ci for cumulative heat
evolution) are different from each other, Eq. (1.2) only holds when all four clinker phases
hydrate at the same rate, which is known to be not true (Hewlett 1998, Escalante-Garcia 1998). If
Portland cement can be produced with simpler compositions (e.g. with only the silicate phases
C3S and C2S), measuring total chemical shrinkage and cumulative heat evolution simultaneously
could potentially allow continuous evaluation of their individual hydration kinetics in the
mixture. The following equations can be derived from Eqs. (6.8) and (6.9) in the absence of C3A
and C4AF,
αC S ( t ) =
3
c2CS ( t ) − a2 H ( t )
pC3S ( a1c2 − a2 c1 )
(6.10)
150
αC S ( t ) =
2
a1 H ( t ) − c1CS ( t )
pC2 S ( a1c2 − a2c1 )
(6.11)
However, due to difficulties of accurately estimating the constants involved in the above
equations, the reliability of the proposed method deserves further investigation.
Despite the inaccuracies, Eq. (1.2) is nevertheless widely used due to its simplicity and
good approximation (early hydration of Portland cement is usually dominated by only two
phases, C3S and C3A). According to Eq. (1.2), the total non-evaporable water content wn(t) and
the cumulative heat evolution H(t) during cement hydration can be related to the total chemical
shrinkage CS(t) by the following equations, respectively,
wn0
CS (t )
CS 0
(6.12)
H0
H (t ) =
CS (t )
CS 0
(6.13)
wn (t ) =
Previous studies have shown that the ratio of CS(t) to wn(t) (evaluated discontinuously) decreases
with increasing curing temperature (Geiker 1983, Zhang 2010). Since both wn0 and H0 are
independent of curing temperature, the results suggest that CS0 decreases with increasing
temperature. As will be shown in the next section, the correlations between CS(t) and H(t) at
different curing temperatures also suggest that CS0 decreases with increasing temperature. Due to
limitations of the temperature control scheme of the chemical shrinkage tests, the test data have
to be calibrated to the same temperature as the corresponding isothermal calorimetry tests by
incorporating a scale factor C into Eq. (6.13), based on the model developed in Chapter 4,
H0
H (t ) =
CS (C ⋅ t )
CS 0
(6.14)
151
where C is related to the temperature of the chemical shrinkage test (TCS) and that of the
isothermal calorimetry test (TIC) by the following equation,
E
C = exp  a
 R

 1
1 
−

 
 TCS TIC  
(6.15)
6.4 Estimating the Correlation Factors
Since the temperatures of the chemical shrinkage tests performed in this study are not
known precisely, the scale factor C cannot be accurately estimated with Eq. (6.15). If, according
to Eq. (6.14), the chemical shrinkage curve is transformed by a scale factor of C, it will be
approximately proportional to the cumulative heat evolution curve with a proportionality
constant of H0/CS0. Therefore, both H0/CS0 and C can be estimated by rescaling the chemical
shrinkage curve based on coordinate transformation roles to achieve the best agreement with the
heat evolution curve. In order to get the best estimation of these constants, both the integral
curves and the derivative curves of hydration kinetics shall be compared. The following equation
can be derived by taking the derivative of Eq. (6.14),
dH (t ) H 0C dCS (C ⋅ t )
=
dt
CS 0
dt
(6.16)
The values of C estimated by the best fit method may be used to derive the temperatures of the
chemical shrinkage tests based on Eq. (6.15). Figures 6.5 shows the heat evolution curves of the
Class H-P cement measured at 25 °C and the transformed chemical shrinkage curves that exhibit
the best agreement. Note that chemical shrinkage test results obtained with thin specimens
(hollow cylinders) are also included for comparison. Some offsets in the time axis are observed
for different tests probably due to variations of the duration of the induction period. The
152
relatively shorter induction period of the chemical shrinkage tests may be attributed to extra
mixing time (using the cement and mortar mixer) adopted for the particular test method (See
Chapter 2). To achieve better agreement between different tests, it is often necessary to offset the
time axis. Figure 6.6 shows that chemical shrinkage agrees almost perfectly with heat evolution
after such offsets except during very early stages (before the acceleration period), where the
hydration rate measured by chemical shrinkage seems to be much higher than that measured by
heat evolution. Since chemical shrinkage (i.e. volume reduction) is likely to be associated with
the formation of hydration products and heat release is likely to be associated with the
dissolution of the anhydrous cement, the discrepancies during the early period are probably
caused by the imbalance between dissolution and precipitation rates as discussed in Section 5.1.
-3
x 10
300
Heat evolution
Transformed from chemical shrinkage (solid cylinders)
Transformed from chemical shrinkage (hollow cylinders)
3
200
2
100
1
0
0
10
20
30
40
Time (h)
50
60
70
Cumulative heat evolution (J/g cement)
Heat evolution rate (W/g cement)
4
0
80
Figure 6.5: Heat evolution curves vs. transformed chemical shrinkage curves (before offset)
(Class H-P cement, w/c = 0.38, 25 °C)
153
-3
x 10
300
Measured heat evolution
Transformed from chemical shrinkage (solid cylinders)
Transformed from chemical shrinkage (hollow cylinders)
3
200
2
100
1
0
0
10
20
30
40
Time (h)
50
60
70
Cumulative heat evolution (J/g cement)
Heat evolution rate (W/g cement)
4
0
80
Figure 6.6: Heat evolution curves vs. transformed chemical shrinkage curves (after offset)
(Class H-P cement, w/c = 0.38, 25 °C)
Figure 6.7 shows the heat evolution curves (derivative curves) of other types of cement
measured at 25 °C and their corresponding best-fit transformed chemical shrinkage curves. For
cements that contain little or no C3A (< 0.3%), such as Class H-P and H-I cement, nearly perfect
agreements can be achieved between the heat evolution curves and the transformed chemical
shrinkage curves. For cements that contain C3A (> 2%), such as Class A, C, and G cement, the
agreements between the two types of curves are less ideal, especially around the hydration peaks.
The differences are mainly due to the fact that the ratios between heat release and chemical
shrinkage associated with different hydration reactions are not constant. It appears that the
second peak (caused by C3A hydration) measured by chemical shrinkage is always higher than
that measured by heat evolution, suggesting that the ratio of chemical shrinkage to heat release
for C3A hydration during this particular period is higher than that for C3S hydration. It should be
154
noted that these ratios are not exactly comparable to the ratios between ai and ci as shown in Eqs.
(3.1) and (6.1), because the hydration of C3A first produces ettringite during the hydration peak,
which then transforms to calcium monosulfoaluminate at later ages (Taylor 1997a, Hewlett
1998). The integral curves of heat evolution of different cements cured at different temperatures
are compared with the transformed chemical shrinkage curves in Figure 6.8. In general, excellent
agreements can be achieved between the two different methods of evaluating cement hydration
kinetics.
Measured heat evolution
Transformed from chemical shrinkage
-3
x 10
4
Class A
3
2
1
0
0
20
Heat evolution rate (W/g cement)
60
3
Class G
2
1
0
x 10
5
4
Class C
3
2
1
0
0
20
20
40
60
3
40
60
Time (h)
-3
x 10
4
0
6
Time (h)
-3
5
40
Heat evolution rate (W/g cement)
5
Heat evolution rate (W/g cement)
Heat evolution rate (W/g cement)
-3
x 10
2
Class H-I
1
0
0
Time (h)
20
40
60
Time (h)
Figure 6.7: Heat evolution curves vs. transformed chemical shrinkage curves
for different types of cement
155
80
Cumulative heat evolution (J/g cement)
80
350
300
250
200
60°C
150
40°C
100
25°C
Class A
50
0
0
20
40
Time (h)
60
350
300
250
200
60°C
150
40°C
25°C
100
Class G
50
0
0
20
40
Time (h)
60
Transformed from chemical shrinkage
Cumulative heat evolution (J/g cement)
Cumulative heat evolution (J/g cement)
Cumulative heat evolution (J/g cement)
Measured heat evolution
400
300
60°C
40°C
200
25°C
100
Class C
0
0
20
40
Time (h)
60
80
300
250
200
60°C
150
40°C
100
25°C
50
0
Class H-P
0
20
40
Time (h)
60
80
Figure 6.8: Heat evolution curves vs. transformed chemical shrinkage curves for different types
of cement at different curing temperatures
Table 6.3 shows the constants used to transform the chemical shrinkage curves in order to
match the heat evolution curves, including the offset time (t0), the scale factor (C) and the
correlation factor (H0/CS0). Since the temperatures of the isothermal calorimetry tests (TIC) are
known precisely, they can be used to estimate the temperatures of the chemical shrinkage tests
using the best-fit scale factors according to Eq. (6.15), i.e.
TCS =
1
R ln C 1
+
Ea
TIC
(6.16)
156
The results are also listed in Table 6.3. The activation energies used for such estimations are the
values obtained by the best fit method for the 25-40 °C temperature range shown in Table 6.2.
The calculated specimen temperatures of the chemical shrinkage tests achieved by the heating
scheme presented in Chapter 2 are slightly higher than the previously estimated values (Table
2.5). For these tests, scale factors less than 1 have to be used to transform the chemical shrinkage
curves to obtain their correlations factors with heat evolution curves. To further demonstrate that
the necessary scale factors are indeed associated with the slightly different temperatures between
chemical shrinkage tests and isothermal calorimetry tests, one additional chemical shrinkage test
(H-I-S) was performed with Class H-I cement by reducing the target temperature set with the
temperature controllers by 5 °F (2.8 °C). As shown in Table 6.3, the result of this particular test
is found to be directly proportional to heat evolution (i.e. C = 1).
Table 6.3: Best-fit parameters and estimated temperatures of chemical shrinkage tests
Test No.
t0 (h)
C
H0/CS0 (J/mL) TIC (°C) TCS (°C)
A-1Sa
0.8
1
7600
25
25.0
A-1
1
1.09
7600
25
23.4
C-1
0.4
0.85
7800
25
28.0
G-1
0.7
0.95
7700
25
25.9
a
H-P-1S
1.8
1
7500
25
25.0
H-P-1
1.2
1.07
7500
25
23.7
H-I-1
1
1.11
8000
25
23.2
A-5
0
0.84
8400
40
43.7
C-5
0
0.9
8350
40
42.1
G-5
0
0.86
8850
40
42.9
H-P-5
0.6
0.9
8700
40
42.2
A-6
0
0.9
9750
60
62.6
C-6
0
0.9
10100
60
62.4
G-6
0
0.91
9750
60
62.0
H-P-6
0
0.85
10200
60
63.8
H-I-Sb
0
1
8850
40
40.0
a
: Supplementary tests with thin specimens (hollow cylinders)
b
: Supplementary tests performed by using a target temperature of
157
At 25 °C, the correlation factor between heat evolution and chemical shrinkage (H0/CS0)
for different cements is found to range from 7500 to 8000 J/mL, well within the previously
reported range of 6500 to 8500 J/mL (Bentz 2008, Bentz 2010). The correlation factor increases
significantly with increasing curing temperature. Since H0 is independent of curing temperature,
the results suggest that CS0 decreases with increasing temperature, consistent with previous
studies (Geiker 1983, Zhang 2010). Figure 6.9 shows the dependence of CS0/H0 on curing
temperature. After normalization, the linear reduction rate of CS0 with increasing temperature is
found to be 0.63%, 0.66%, 0.59%, 0.75%, and 0.64% per °C for Class A, C, G, H-P, and H-I
cement, respectively. Note that the slope for Class H-I cement was determined with only two
data points since chemical shrinkage was not obtained at 60 °C. The results are in reasonable
agreement with the reduction rate of 0.783% per °C used in Section 3.6.
0.15
Class
Class
Class
Class
Class
0.14
CS0/H0 mL/kJ
0.13
A
C
G
H-P
H-I
0.12
0.11
0.1
0.09
20
25
30
35
40
45
50
Curing temperature (°C)
55
60
65
Figure 6.9: Dependence of correlation factor (CS0/H0) on curing temperature
158
6.5 Summary
The traditional isothermal calorimetry tests are used to further evaluate the hydration
kinetics of different types of cement used in this study. The one-parameter model developed in
Chapter 4 to model the effect of curing temperature on cement hydration kinetics is further
verified here with the heat evolution data obtained from isothermal calorimetry tests. Due to the
fact that the different compounds in cement hydrate at different rates and have different
temperature sensitivities, the model is found to be more accurate for cement with simpler
compositions (e.g. no C3A content) and for smaller temperature changes (e.g. < 15 °C). A new
method of estimating the apparent activation energies of Portland cement is proposed. The
activation energies of Class A, C, G, H-P and H-I cements determined for the temperature range
of 25-60 °C are 40.2, 37, 40.2, 35.5, and 37 kJ/mol, respectively.
The reliability of using chemical shrinkage test results to measure cement hydration
kinetics is evaluated by studying its correlations with heat evolutions during the hydration
process. After some calibrations, total chemical shrinkage is found to be almost perfectly
proportional to cumulative heat evolution at all curing temperatures investigated in this study,
especially for cement with negligible amount of C3A. The proportionality constant, namely the
ratio of total chemical shrinkage to total heat release at complete hydration (CS0/H0), is found to
vary slightly with cement composition and decrease significantly with increasing curing
temperature. While H0 is independent of curing temperature, CS0 decreases approximately
linearly with increasing temperature at a rate of 0.59% to 0.75% per °C for different cements.
159
CHAPTER 7 : EFFECT OF CURING TEMPERATURE AND
PRESSURE ON THE PHYSICAL AND MECHANICAL
PROPERTIES OF CEMENT
7.1 Introduction
Cement paste develops its properties by hydration of cement to form a complex series of
hydrates that bind cement grains together to form a continuous matrix. The most dramatic
property change that a cement paste experiences during hydration is the solidification process
known as setting, which transforms the paste from a workable plastic slurry into a rigid solid. In
oilwell cementing, the property of a plastic cement paste is typically quantified by its viscosity
while the property of a hardened cement paste may be quantified by its permeability,
compressive strength, tensile strength, modulus of elasticity, Poison’s ratio, etc. As a result of the
progressive cement hydration reactions, these various physical and mechanical properties also
vary with time. The overall degree of hydration of cement, which is quantifiable during the entire
hydration process, can be used to model the transition from the plastic state to the solid state as
well as the progress of their various properties with time. From the test results shown in the
previous chapters, it has become clear that the effect of curing temperature and pressure on
cement hydration kinetics (i.e. degree of hydration vs. time) can be explained by chemical
kinetics theories, because cement hydration is essentially a complex chemical process. However,
unlike the overall degree of hydration, which is only related to the total amount of cement
reacted or the total amount of hydration products generated during hydration, the physical and
mechanical properties are also affected by the arrangement and the shape of the hydration
160
products, namely the microstructure of the material, and hence are much more difficult to model,
especially at later ages.
Theoretically, if some physical or mechanical properties (represented by I) of a given
cement paste, such as viscosity, permeability, strength, and modulus of elasticity, can be
uniquely related to the degree of hydration of cement, e.g. I = h(α), then the effect of curing
temperature and pressure on the time evolution of these properties can be modeled in the same
way as hydration kinetics (Chapter 4), i.e.:
I r = h (α r ) = h ( f ( t r ) ) = F ( t r )
(
)
I = h (α ) = h f ( C ( T , P ) ⋅ t ) = F ( C ( T , P ) ⋅ t )
(7.1)
(7.2)
where Eq. (7.1) represents the time evolution of a particular property at the reference condition
(Tr, Pr) while Eq. (7.2) represents the time evolution of this property at any curing condition (T,
P). The scale factor C(T, P) is the same as defined in Eq. (4.7). The concept is similar to the
maturity method used to estimate concrete strength (ASTM C1074 2010, Carino 1991, Carino
2001, Schindler 2004, Viviani 2005). Clearly, the validity of this model depends on the existence
of a one-to-one relationship between the particular property and the degree of hydration
(maturity) of cement. While most studies have proven that early-age properties such as viscosity,
setting time and early-age compressive strength are generally uniquely related to the degree of
hydration (Scherer 2010, Pinto 1999, García 2008, Zhang 2010, Kjellsen 1991), other studies
have found poor correlations between strength and maturity, particularly at later ages (Klieger
1958, Alexander 1962, Volz 1981, Carino 1983, Kjellsen 1993). These relatively poor
correlations are primarily due to the effect of curing temperature on the microstructure
development of cement pastes (Kjellsen 1990, Kjellsen 1991).
161
For verification of the model represented by Eqs. (7.1) and (7.2), test programs completely
different from those adopted in this study are needed as it requires the relevant physical or
mechanical properties to be measured continuously or at a reasonable number of different ages
under different curing conditions. Since there is not yet an established destructive test method to
determine the mechanical properties of oilwell cement under in-situ conditions of high
temperature and pressure, the pressure cell test series are mainly used in this chapter to validate
the innovative tensile strength test method developed in this study and to study the potential
specimen damage mechanisms during depressurization. Some qualitative analyses of the effect
of curing temperature and pressure on the density and tensile strength of the hardened cement are
also performed.
7.2 Results of the Preliminary Tests
Since the water pressure test is not a well established tensile strength test method, it is
important to compare the test results with those of the traditional tests, such as the splitting
tensile tests. Unfortunately, due to the small specimen diameter, splitting tensile strength is found
to be strongly dependent on several influencing factors. The preliminary test results reported here
are used mainly to establish standard test parameters for the splitting tensile tests such that
reasonable results can be obtained. The cylinder specimens of Tests PL-1, PL-2, and PL-3 (see
Table 2.3) were tested with plywood bearing strips while those of Tests PL-4 and PL-5 were
tested with both plywood and basswood. The type of wood and thickness of the bearing strip
used were found to have negligible effects on test results. On the other hand, the width of the
bearing strip was found to have a significant impact on test results. In order to directly compare
results of different tests, which were obtained not at the exact same age and curing conditions,
162
the average splitting tensile strength of each cylinder was normalized by the average tensile
strength of all briquette specimens (i.e. direct tensile strength) of the batch. Since no briquette
was made for Test PL-1, the splitting tensile strength obtained without bearing strips was
assumed to be approximately equal to the direct tensile strength as supported by the test results
of Test PL-2. The effect of the relative bearing strip width (b/D) on the average normalized
splitting tensile strength of cylinders of different height is presented in Figure 7.1. The error bars
represent the standard deviations of test results. The splitting strength to direct tensile strength
ratio ranged from 0.56 to 1.61 when the b/D ratio varied from 0.04 to 0.16. Such dependence is
due to the fact that the width of the bearing strips controls the stress intensity that a specimen is
subjected to by changing the size of loading area. The splitting strength seems to reach a peak
when the b/D ratio equals 0.16 as further increasing the bearing strip width will not change the
contact area between the specimen and the bearing strip. For the 178mm high cylinders, a b/D
ratio between 0.08 and 0.09 seems to be optimal since it gives a splitting tensile strength closest
to the direct tensile strength.
Splitting strength/direct tensile
strength
height=102mm
height=178mm
height=305mm
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.04
0.08
0.09
0.16
Relative bearing strip width (b/D)
0.5
Figure 7.1: Effect of bearing strip width on splitting tensile strength
163
Due to settling, the upper section of a vertically cast cylindrical specimen usually has a
lower density and strength than the lower section. Figure 7.2 shows the effect of such settling on
the splitting tensile strength of cylinder segments taken from different regions of the specimens.
The five segments cut from a 305mm cylinder specimen were marked from top to bottom as:
topmost, top, middle, bottom, and bottommost, respectively. The segments cut from shorter
cylinders are self-explanatory in the figure. In order to include all samples tested with different
b/D ratios, the strengths of samples taken from different regions of the cylinder were normalized
by the samples’ average strength. The ratio of the sample strength to average strength seemed
independent of the bearing strip width but varied significantly with the vertical position of the
sample. The strength of the bottommost sample was found to be 17 to 44% higher than that of
the topmost one, depending on the height of the cylinder, while the strength of the middle section
is most representative of the average strength of the sample. Therefore, for the pressure cell test
series, splitting tensile tests were performed on the middle section of the specimens with a
relative bearing strip width of approximately 0.08.
height=102mm
height=178mm
height=305mm
1.4
Strength ratio
1.2
1
0.8
0.6
0.4
0.2
0
Topmost
Top
Middle
Bottom Bot.-most
Sample position
Figure 7.2: Splitting tensile strength variation along vertical direction of cylindrical samples
164
7.3 Results of the In-Situ Pressure Cell Tests (Series I)
Because of settling and water filtration through the filter paper, the final solid weight of a
specimen cast in the pressure cell was smaller than its initial liquid weight. The average weight
losses were found to be 1.9, 1.1, and 0.6% for slurries (Class H-II cement) cured at 24, 40.6 and
60 °C, respectively. Since the cement contents of the slurries remain constant, the final effective
w/c ratios of the hardened cement paste can be calculated from the weight losses and are
determined to be 0.373, 0.385, and 0.391 for curing temperatures of 24, 40.6 and 60 °C,
respectively. The results suggest that the loss of water due to settling and filtration is reduced
significantly at higher temperatures thanks to faster setting of cement. The weight losses would
have been greater if the amount of water imbibed during the curing period (approximately 0.8%
by weight) was subtracted from the final weight. Figure 7.3 shows typical locations of fracture
planes of in-situ water pressure tests. It is observed that occasionally extra fracture planes might
occur along the active seals besides the one between the seals due to high seal pressure (13.8
MPa). Because of the relatively high w/c ratio used in this test series, which causes severe
settling, hydraulic fracture typically occurs in the upper section of the specimens.
Figure 7.3: Typical locations of fracture planes under hydraulic pressure
165
Figure 7.4 shows both the in-situ water-pressure tensile strength (ffpt) and the splitting
tensile strength (fst) test results obtained at ambient temperatures. The coefficient of variation of
different test sets ranges between 0.03 and 0.23. No significant differences are observed between
the two different batches of cement (i.e. Class H-I and H-II). As a reminder, fst was determined
after the specimen had failed in a pressure cell. As will be shown in Section 7.4.3, water pressure
change does not seem to cause damage to specimens produced with Class H cement probably
due to their relatively high permeability. Therefore, in this test series, specimens used for
splitting tests were probably not damaged. Splitting tensile strength seems to increase slightly
with increasing curing pressure. However, the amount of increase is found to be very small, often
negligible. The maximum increases for Class H-I and H-II cements were 19% and 6%,
respectively. Note that test results are also affected by lab temperature fluctuations. For both test
procedures, A and B, water pressure tensile strength increases relatively more significantly (17 to
32%) with increasing curing pressure. It should be noted that there exist not yet an official
definition of the in-situ tensile strength of oilwell cement. The in-situ water pressure tensile
strength presented in this study is likely to be affected by factors other than the material
properties. For example, although using different test methods, Clayton (1980) has found that the
stress difference (between the annulus water pressure and the axial stress) required for a concrete
specimen to fracture increased with increasing axial stress. Since the axial pressure of the
specimen increases with increasing curing pressure, it is natural for water pressure tensile
strength (pressure difference between the annulus and the ends) to increase with increasing
curing pressure. For specimens cured under the same pressure (Test sets 24-II and 24-III),
Procedure B seems to give slightly lower test results than Procedure A probably due to the lower
axial pressures during the tests.
166
Tensile strength (MPa)
5
Splitting (H-I)
Splitting (H-II)
Water pressure (H-I)
Water pressure (H-II)
Procedure A
Procedure B
4
3
2
1
0
0.69
6.9
6.9
Curing pressure (MPa)
13.1
Figure 7.4: Comparison between test results of Class H-I and H-II cements
Figure 7.5 shows the splitting tensile test results of Class H-II cement at different curing
conditions. For the range investigated in this study, the splitting tensile strength is found to
increase very little (less than 11%) with increasing curing pressure. However, it increases
dramatically, approximately 25% and 60%, when curing temperature is increased from 24 °C to
40.6 °C and 60 °C, respectively. These results are consistent with previous hydration kinetics
analyses (Chapters 4 and 5) that found hydration rates are more sensitive to temperature changes
than pressure changes. Figure 7.6 shows the in-situ water pressure test results of Class H-II
cement at different curing conditions. Most test results appear to vary randomly between 3 and 4
MPa, indicating that neither curing temperature nor curing pressure (within the ranges studied)
has a significant effect on water pressure tensile strength. As discussed in Section 1.2.3, the fluid
pressure tensile strength depends on both fluid viscosity and specimen permeability. Since the
viscosity of water decreases with increasing temperature (Korson 1969, Viswanath 2007) and the
permeability of cement paste increases with increasing curing temperature (Goto 1981), the
167
water pressure test results obtained at different temperatures probably should not be directly
compared with each other. More comprehensive test programs are needed to correct the test
results for the effect of fluid viscosity and specimen permeability or an alternative fluid medium
Splitting tensile strength (MPa)
should be used in the future.
4.5
4
3.5
2.5
Curing
Temperature
Amb.
2
40.6°C
3
1.5
60°C
1
0.5
0
0.69
6.9
13.1
Curing pressure (MPa)
Figure 7.5: Effect of curing condition on splitting tensile test results
(Class H-II cement, w/c = 0.4, age = 48 h)
Water pressure tensile strength
(MPa)
5
Procedure A
4
Curing
Temperature
Amb.
3
Procedure B
2
40.6 °C
60 °C
1
0
0.69
6.9
6.9
13.1
Curing pressure (MPa)
Figure 7.6: Effect of curing condition on in-situ water pressure test results
(Class H-II cement, w/c = 0.4, age = 48 h)
168
The water pressure tensile strength is compared with splitting tensile strength in Figure 7.7
for Class H-II cement at different curing temperatures. Note that test results of all curing
pressures are included. The correlation factor (α = ffpt/fst) between water pressure tensile strength
and splitting tensile strength was found to be 1.36, 1.16, and 0.89 for tests conducted at curing
temperatures of 24, 40.6, and 60 °C, respectively. The results suggest that the correlation factor
decreases with decreasing fluid viscosity and increasing specimen permeability, which is
consistent with our analysis in Section 1.2.3. Sometimes defective specimens with large air voids
near the top of the filter papers were produced at high curing temperatures, which might be
caused by steam generated from the wet filter papers during the preheating period. Hydraulic
fractures of defective specimens typically occurred in planes containing these voids. These
observations may explain why the correlation factor at 60 °C is slightly less than expected (< 1).
The results of Series II of pressure cell tests suggest that the problem can be avoided if
temperature was raised after the specimens had been cast.
Tensile strength (MPa)
Splitting
Water pressure
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
24
40.6
60
Curing temperature (°C)
Figure 7.7: Average splitting tensile strength vs. average water-pressure tensile strength
(Class H-II cement, w/c = 0.4, age = 48 h)
169
7.4 Results of the Non In-Situ Pressure Cell Tests (Series II)
7.4.1 Effect of curing conditions on the physical properties of cement
Unless the w/c ratio is properly tuned or certain additives are used, settling and bleeding of
cement pastes before setting typically cannot be completely avoided. After setting, the cement
paste imbibes water from the surrounding environment due to chemical shrinkage. Therefore, the
total water content of a cement paste does not remain constant during the process of hydration.
Since both setting time and chemical shrinkage depend strongly on curing condition, the total
water content at a given time also varies with the curing condition. As discussed in the previous
section, the final water content of a hardened cement paste (and the effective w/c ratio) can be
determined by measuring its weight changes. Table 7.1 shows the initial w/c ratios of the
different slurries used in this test series as well as the final w/c ratios of the specimens after three
days of curing at different conditions. Except for the slurries prepared with Class C cement at a
w/c ratio of 0.56 and those prepared with Class H-II cement at a w/c ratio of 0.3, all other slurries
showed a net loss of water. The total amount of water lost appears to depend strongly on curing
temperature but seems independent of curing pressure, consistent with the test results obtained in
Section 7.3 for Class H-II cement slurries with a w/c ratio of 0.4. The density of a hardened
cement paste also varies with curing temperature because it is directly related to the final
effective w/c ratio. In addition, due to settlement of cement particles, the density (and effective
w/c ratio) of a specimen is not uniform along the vertical direction. After removal from the
pressure cells, each hardened specimen was separated into three segments along the vertical
direction (top, middle and bottom) to obtain their specific gravities and densities using the
Archimedes Principle (API 1997). Separation of the three different segments was not exactly the
same for different specimens due to variations of the location of the fracture plane.
170
Table 7.1: Physical properties of specimens from different tests
w/c
Test No.
A-1
A-2
A-3
A-4
A-5
A-6
C-1
C-2
C-2
C-3
C-4
C-5
C-6
G-1
G-2
G-3
G-4
G-5
G-6
H-P-1
H-P-2
H-P-3
H-P-4
H-P-5
H-P-6
H-II-5-1
H-II-5-2
H-II-5-3
H-II-1
H-II-2
H-II-3
H-II-4
H-II-3-1
H-II-3-2
H-II-3-3
Temp. Press.
(°C) (MPa) Slurry
24.4
22.8
25
24.4
40.6
60
26.9
27.5
25.6
25
25.6
40.6
60
25
24.7
23.1
25
40.6
60
25.6
22.2
23.9
26.1
40.6
60
26.1
26.1
26.7
-
0.69
17.2
34.5
51.7
0.69
0.69
0.69
17.2
17.2
34.5
51.7
0.69
0.69
0.69
17.2
34.5
51.7
0.69
0.69
0.69
17.2
34.5
51.7
0.69
0.69
0.69
17.2
34.5
0.69
17.2
34.5
51.7
0.69
17.2
34.5
0.46
0.56
0.44
0.38
0.5
0.38
0.3
Solid
0.422
0.416
0.432
0.427
0.436
0.444
0.581
0.581
0.583
0.576
0.579
0.582
0.573
0.401
0.395
0.393
0.402
0.408
0.408
0.354
0.350
0.351
0.359
0.362
0.362
0.432
0.429
0.424
0.371
0.370
0.364
0.369
0.305
0.306
0.301
Slurry
1.904
1.805
1.944
2.020
1.837
Density (g/cm3)
Solid
Top
Mid.
1.960 2.038
1.963 2.060
1.975 2.035
1.984 2.048
1.946 2.015
1.931 1.995
1.839 1.866
1.841 1.873
1.838 1.871
1.845 1.876
1.848 1.880
1.828 1.862
1.824 1.859
2.002
2.023 2.102
2.028 2.116
2.016 2.093
1.988 2.068
1.992 2.061
2.086 2.160
2.085 2.160
2.092 2.146
2.068 2.127
2.063 2.129
-
Bot.
2.054
2.064
2.043
2.062
2.034
2.015
1.884
1.883
1.887
1.890
1.895
1.879
1.877
2.106
2.109
2.113
2.094
2.089
2.072
2.173
2.167
2.160
2.156
2.141
2.024
2.226
-
Settlement
(%)
10.6
10.2
10.4
10.2
8.4
7.0
2.8
2.5
3.5
3.9
2.2
2.5
2.6
9.6
11.2
10.2
9.2
9.5
7.4
7.9
7.4
5.8
5.9
6.2
13.8
10.8
11.8
4.8
5.2
6.0
5.1
1.3
1.5
171
For comparison purpose, a Halliburton fluid density scale was used to measure the slurry
densities in the plastic state (without pressurization). Total settlement during hydration was
obtained by dividing the height change of the specimen (i.e. the distance between the top surface
of the hardened specimen and the top of the rubber sleeve) by its original height. The average
results of these tests are listed in Table 7.1. Densities of the hardened specimens were higher
than those of the slurries due to chemical shrinkage and settling effect. Total settlement was
found to be largely independent of curing conditions, except for slurries with relatively serious
water losses, for which high curing temperature tended to reduce settlement. It should be noted
that measurements of total settlement were only approximate due to the fact that specimen
surfaces are usually not flat.
Figure 7.8 shows the average segment densities of Class A cement specimens (w/c = 0.46)
at different curing conditions. Probably due to the dilution effects, the top segments of the
specimens are found to have much lower densities than the middle and bottom segments.
Therefore, the measured densities of the top segments were more strongly affected by their sizes
and hence less reliable for comparisons between different tests. The specimen density (middle
and bottom segments) appears to decrease with increasing curing temperature, which is
consistent with water loss calculations since lower density corresponds with smaller water loss
(i.e. higher final w/c ratio). Curing pressure does not appear to have a significant effect on
specimen density. The density test results of both the slurries and the hardened specimens are
plotted against their w/c ratios in Figure 7.9. Both the slurry density and the solid density are
found to vary approximately linearly with w/c ratio, regardless of the cement type and the curing
condition. The results suggest that the absolute densities of different types of cement are more or
less the same and that the degree of hydration has a relatively small effect on specimen density.
172
Average section density (g/cm3)
Top
Middle
Bottom
2.1
2.05
2
1.95
1.9
1.85
60°C
40.6°C 24.4°C 22.8°C
25°C
24.4°C
0.69MPa 0.69MPa 0.69MPa 17.2MPa 34.5MPa 51.7MPa
Curing condition
Figure 7.8: Dependence of specimen density on curing condition
(Class A cement, w/c = 0.46, age = 72 h)
Slurry
Solid-Middle
Solid-Bottom
2.20
Average density (g/cm3)
2.15
2.10
2.05
2.00
1.95
1.90
1.85
1.80
1.75
0.30
0.35
0.40
0.45
0.50
0.55
0.60
Effective w/c ratio
Figure 7.9: Dependence of specimen density on effective w/c ratio of different cement
173
7.4.2 Deformation behavior of cement paste during pressurization and depressurization
According to the definition of the bulk modulus (Eq. (2.1)), for a cement paste with a
volume of V and a bulk modulus of K under hydrostatic pressure P, the infinitesimal change in
its volume is proportional to the infinitesimal hydrostatic pressure change, i.e.
dV = −
V
dP
K
(7.3)
However, the deformation of the cement paste has to be measured together with a pressurizing
medium, such as water. Theoretically, for a system that only consists of cement, mixing water
and pressurizing water with known volumes for each, the bulk modulus of cement particles can
be estimated by measuring the volume change of the entire system as a function of pressure.
V
V 
dVS = dVC + dVW = −  C + W  dP
 KC KW 
(7.4)
where the subscripts S, C, and W represent the entire system, total cement, and total water,
respectively. In reality it is very difficult to obtain reliable results because only a small fraction
of the measured total volume change is due to the deformation of cement particles (the bulk
modulus of cement is about 50 times higher than that of water (Lin 2006)). The majority of the
total volume change is due to the deformation of water. Obviously, Eq. (7.4) is only applicable
when the hydration of cement is negligible. As hydration progresses, part of the free water
(capillary water) reacts with cement to become non-evaporable water and gel water. As
discussed in Chapter 3, although gel water may be assumed to have similar properties as free
water, non-evaporable water has a much higher density (and possibly bulk modulus). Therefore,
the deformation behavior of a system consisting of a cement paste and pressurizing water varies
with time due to cement hydration.
174
The deformation of a Class H-I cement paste (with a w/c ratio of 0.38) cured at 34.5 MPa
and ambient temperature was studied by depressurization and re-pressurization at six different
ages (1, 6, 12, 24, 48, and 72 hours). The pressure gradient used for early-age tests (1, 6, 12, and
24 hours) was 3.45 MPa/min while that used for older-age tests (48 and 72 hours) was 0.345
MPa/min. The pressure and total volume of the entire system are plotted as a function of time in
Figure 7.10. Apparently, the total volume of the system gradually decreases with time as a result
of chemical shrinkage. It should be noted that it is very difficult to accurately estimate the actual
total volume of the entire system due to the “dead volume” mentioned in Chapter 2. However,
System pressure (MPa)
this does not prevent us from performing some qualitative comparisons between test results.
40
30
20
10
0
0
10
20
30
40
Time (h)
50
60
70
80
0
10
20
30
40
Time (h)
50
60
70
80
Total system volume (mL)
1800
1780
1760
1740
1720
1700
Figure 7.10: Pressure and volume variations with time of a system mainly consists of cement
paste and pressurizing water
175
Figure 7.11 shows the depressurization and re-pressurization curves (total volume vs.
pressure) of the system at different test ages. It is obvious that the unloading curves agree very
well with the loading curves for all test ages. The bulk modulus of the entire system can be
calculated from these curves based on Eq. (2.1). The results obtained at three different test ages
are shown in Figure 7.12. As discussed in Section 2.4.2.1, the seemingly high compressibility of
the system (small bulk modulus) at very low pressures (< 3 MPa) is probably due to the effect of
the entrapped air. The bulk modulus derived from the unloading curves is found to be slightly
higher than those derived from the loading curves at relatively high pressures. The causes of such
divergences are still not clearly understood mainly because calculations of the bulk modulus is
based on the total deformations of the entire system including the cement paste, pressurizing
water, rubber sleeves, active seals, and the containers (pressure cells and Syringe Pump A).
1800
1790
1h
Total system volume (mL)
1780
Depressurization
Re-pressurization
6h
1770
12h
1760
1750
24h
1740
48h
1730
72h
1720
1710
0
5
10
15
20
Pressure (MPa)
25
30
35
Figure 7.11: Variation of total system volume with pressure at different ages
176
4
Depressurization
System bulk modulus (GPa)
3.5
72h
3
24h
2.5
Re-pressurization
1h
2
1.5
0
5
10
15
20
Pressure (MPa)
25
30
35
Figure 7.12: Variation of system bulk modulus with pressure
Since the total volume of the system changes very little (about 1%) during the loading and
unloading processes, the deformation gradient of the system is approximately inversely
proportional the bulk modulus of the system, i.e.
dVS
V
=− S
dP
KS
(7.5)
It is obvious that the deformation gradient is proportional to the water flow rate when a constant
pressure gradient is adopted. Figure 7.13 shows the deformation gradient of the system at two
different test ages. For the depressurization curve at the age of 72 hours, a small period of rapid
increase in system deformation gradient (flow rate) is observed before reaching the steady state.
Since the compressed pore water inside a hardened cement paste needs to flow out during
depressurization due to volume expansion, the pore pressure is not likely to be uniformly equal
to the surrounding water pressure during the dynamic process. The initial rapid increase in flow
rate is probably caused by the increasing pressure difference between pore pressure and the
177
surrounding water pressure and this rapid increase period probably ends when the increased
pressure difference is sufficient to enable a steady state during which the flow rate can keep up
with the volume expansion of the pore water. This threshold pressure difference is likely to
increase with decreasing permeability of the cement paste, and may cause damage to specimens
when permeability is too low. The subsequent gradual increase in flow rate is due to decreasing
water modulus with decreasing pressure and the final rapid increase at very low pressures is
probably due to the release of entrapped air. In contrast, during re-pressurization with a constant
pressure gradient, the flow rate first decreases rapidly due to compression of the entrapped air
and then decreases gradually due to increasing water bulk modulus. However, the rapid decrease
of flow rate is not observed when re-pressurization of a hardened cement paste immediately
follows depressurization. This is probably due to the fact that the system was not given enough
time to reach a balanced state. Other test results have shown that total system volume continues
to increase for a period of time after it has been depressurized.
System deformation gradient (mL/MPa)
-0.45
Depressurization
-0.5
-0.55
-0.6
72h
-0.65
-0.7
Re-pressurization
1h
-0.75
-0.8
-0.85
0
5
10
15
20
Pressure (MPa)
25
30
35
Figure 7.13: Variation of system deformation gradient with pressure
178
7.4.3 Effect of curing condition on the mechanical properties of cement
The mechanical properties of a cement paste are directly related to its microstructure. Since
this is determined by many factors, such as w/c ratio, degree of hydration, and the morphology of
hydration products, all of which are affected by the curing condition, it is very difficult to
accurately model the dependence of mechanical property on curing condition. In this study, the
mechanical properties of cement pastes are mainly characterized by splitting tensile strength and
water pressure tensile strength. It is even more challenging to model the water pressure test
results since they are strongly affected by the viscosity of the water and the permeability of the
specimen as discussed in Section 1.2.3. A much more comprehensive test program is required to
effectively quantify the various influencing factors of the tensile strength test results. Only
qualitative analyses of the test results can be performed in this study.
As mentioned in Section 2.3.2, due to limitations of equipment capacities, all the specimens
produced in this test series were depressurized after approximately 72 hours of curing before the
mechanical tests (i.e. not tested at in-situ conditions). Therefore, it is important to study potential
damages to the specimens caused by depressurization. As will be shown in the following
discussions, it was found that the existence of pre-existing defects in specimens due to
depressurization can be determined by both investigating their fracture planes and analyzing the
deformation behavior of the system during the depressurization process. A potential damage
mechanism is proposed in this section based on test results of this study.
7.4.3.1 Effect of curing temperature on tensile strength of cement
Figure 7.14 shows the splitting tensile strength test results of different types of cement at
the age of 72 hours cured under different temperatures and a constant curing pressure of 0.69
MPa. With a coefficient of variation ranging between 0.05 and 0.26, the splitting tensile strength
179
is found to increase 29 to 37% when curing temperature was increased from ambient to 60 °C.
The percentage increase is much smaller than that obtained at the age of 48 hours (Figure 7.5)
mainly because the effect of curing temperature on the total degree of hydration is more
significant at early ages than at later ages (Figure 4.14). In some cases, a slight decrease in
splitting tensile strength is observed when curing temperature was increased from ambient to
40.6 °C, probably due to the increase in w/c ratio as a result of reduced bleeding (Table 7.1).
Figure 7.15 shows the water pressure tensile strength of the same specimens. The coefficient of
variation is found to vary between 0.03 and 0.26. Test results of Class A and C cements show
that the strength decreases with increasing curing temperature, while test results of Class G and
H-P cements show that the strength is largely independent of curing temperature. The
inconclusiveness of test results further suggests that water pressure test results should not be
directly compared with each other due to the differences in the viscosity of water and the
Splitting tensile strength (MPa)
permeability of specimens as discussed in Section 7.3.
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Amb.
40.6 °C
60 °C
Class A
Class C
Class G Class H-P
Type of cement
Figure 7.14: Effect of curing temperature on the splitting tensile strength of cement
(age = 72 h, curing pressure = 0.69 MPa)
Water pressure tensile strength (Mpa)
180
8
7
6
5
4
Amb.
3
40.6 °C
60 °C
2
1
0
Class A
Class C
Class G
Class H-P
Type of cement
Figure 7.15: Effect of curing temperature on the water pressure tensile strength of cement
(age = 72 h, curing pressure = 0.69 MPa)
7.4.3.2 Effect of curing pressure on tensile strength of cement
Since the degree of hydration is one of the most important factors influencing of the
mechanical properties of cement, it is necessary to revisit the hydration kinetics test results
before investigating the effect of curing pressure on the tensile strength of cement. Figure 7.16
shows the effect of curing pressure on the hydration kinetics of different types of cement
obtained at ambient temperatures. The result of Test H-I-2 was not included because the test was
terminated at the age of 24 hours due to a leakage problem. The final degree of hydration at the
end of the curing period was found to increase 6 to 15% for a curing pressure increase of 50 MPa.
However, for a curing pressure increase of 16.5 MPa, the change in final degree of hydration was
found to vary between -5 and 12%. The large variations in test results are mainly caused by lab
temperature fluctuations. Therefore, one can also expect the tensile strength test results to have
some random variations due to the different lab temperatures.
181
0.69MPa
17.2MPa
Degree of hydration
0.8
51.7MPa
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
Class C
Class A
0.2
0
Degree of hydration
34.5MPa
1
0.2
0
20
40
Time (h)
0
60
0
0
20
40
Time (h)
60
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0
0.2
Class H-P
0
20
40
Time (h)
60
0
20
40
Time (h)
0
20
0.2
Class H-I
0
Class G
0.2
60
0
40
Time (h)
60
Class H-II
0
20
40
Time (h)
60
Figure 7.16: Effect of curing pressure on the hydration kinetics of different cements
(ambient curing temperatures)
7.4.3.2.1 Class H-I and H-II cements
All specimens prepared with Class H-I and H-II cements, regardless of w/c ratio and curing
pressure, were found to have clean, flat fracture planes during both water pressure tests and
splitting tensile tests. Figure 7.17 and Figure 7.18 show pictures of fractured specimens after
water pressure tests and splitting tensile tests, respectively. Note that only one representative
specimen from each test is shown in Figure 7.17. Figure 7.19 shows the pictures of all specimens
of Test H-II-4 as well as more detailed pictures of the fracture planes. The absence of significant
differences between the fracture patterns of specimens cured under different pressures suggests
182
that depressurization at the rate of 0.345 MPa/min is not likely to cause damage to cement pastes
prepared with Class H-I and H-II cements at the age of 72 hours. However, the hydraulic fracture
planes were found to be primarily located in the upper sections of the specimens for high w/c
ratio (0.5) but tended to be more randomly located for relatively low w/c ratios (0.38 and 0.3),
primarily due to the fact that the severity of settling increases with increasing w/c ratio.
Figure 7.17: Representative fractured specimens after water pressure tests (Class H-II cement)
183
Figure 7.18: Fractured specimens after splitting tensile tests (Class H-II cement)
Figure 7.19: All specimens of Test H-II-4 after water pressure tests
184
The system deformation behaviors during depressurization of tests conducted with Class HI and H-II cements (for a uniform w/c ratio of 0.38) are presented in Figure 7.20. Figure 7.21
shows the test results of Class H-II cement pastes with different w/c ratios. For cement pastes
with the same composition (same cement and w/c ratio), the measured system deformation
gradients converge reasonably well despite their different curing pressures. The small differences
between different tests may be attributed to the fact that the cement pastes achieved slightly
different degrees of hydration and that the total amount of cement paste and pressurizing water
were not exactly the same for different tests. The system deformation was found to follow a
similar behavior as discussed in Section 7.4.2. The deformation gradient (absolute value)
increased with increasing w/c ratio due to the fact that the bulk modulus of water is significantly
lower than that of cement.
System deformation gradient (mL/MPa)
-0.4
-0.45
-0.5
-0.55
H-I-3
H-I-4
H-II-2
H-II-3
H-II-4
-0.6
-0.65
-0.7
0
10
20
30
40
Pressure (MPa)
50
60
Figure 7.20: Variation of system deformation gradient with pressure
(Class H-I and H-II cements, w/c = 0.38)
185
System deformation gradient (mL/MPa)
-0.4
-0.45
-0.5
-0.55
-0.6
-0.65
H-II-3-3, w/c = 0.3
H-II-3, w/c = 0.38
H-II-5-3, w/c = 0.5
H-II-5-2, w/c = 0.5
-0.7
-0.75
-0.8
0
5
10
15
20
Pressure (MPa)
25
30
35
Figure 7.21: Variation of system deformation gradient with pressure (Class H-II cement)
Figure 7.22 shows the tensile strength test results of Class H-I and H-II cement pastes with
a uniform w/c ratio of 0.38 for different curing pressures. Note that Test H-I-2 (17.2 MPa) failed
due to a leakage problem. Similar to the in-situ tests in the first test series, no significant
differences were observed between the tensile strengths of the two different batches of cement
(i.e. Class H-I and H-II). The coefficient of variation ranges between 0.05 and 0.20 for splitting
tensile tests and between 0.04 and 0.10 for water pressure tests. The splitting tensile strength
appears to increase slightly with increasing curing pressure (approximately 10% for a pressure
increase of 50MPa). However, the splitting tensile test results of Class H-I cement also seem to
show some random variations probably due to lab temperature fluctuations. The water-pressure
tensile strength increased relatively more significantly with increasing curing pressure
(approximately 30% for a pressure increase of 50MPa), probably due to the combined effect of
increasing degree of hydration and decreasing permeability of the specimen.
186
Splitting (H-I)
Splitting (H-II)
Water pressure (H-I)
Water pressure (H-II)
Tensile strength (MPa)
6
5
4
3
2
1
0
0.69
17.2
34.5
51.7
Curing pressure (MPa)
Figure 7.22: Effect of curing pressure on the tensile strength of Class H-I and H-II cement
(w/c = 0.38, age = 72 h)
Figure 7.23 shows the tensile strength test results of Class H-II cement pastes with w/c
ratios of 0.3 and 0.5 for different curing pressures. The coefficient of variation ranges between
0.05 and 0.21 for splitting tensile tests and between 0.01 and 0.18 for water pressure tests. Both
the water-pressure tensile strength and the splitting tensile strength were found to increase
slightly with increasing curing pressure with some random variations possibly due to lab
temperature fluctuations. Therefore, to better quantify the effect of curing pressure on the tensile
strength of cement pastes at early ages, it is crucial to exercise more precise temperature control
during the curing process.
The correlation factor between water pressure tensile strength and splitting tensile strength
(α = ffpt/fst) was found to vary between 1.28 and 1.78 for different tests and seemed to be
independent of w/c ratio. The large variation was probably due to the fact that results of both test
187
methods were affected by many influencing factors. The average correlation factor (at the age of
72 hours) was determined to be 1.54±0.16, slightly higher than that determined from the in-situ
tests (1.36) at the age of 48 hours, probably due to decreased specimen permeability with
increasing test age (water pressure test results typically increases with decreasing specimen
permeability).
6
Splitting
(w/c=0.3)
Tensile strength (Mpa)
5
4
Splitting
(w/c=0.5)
3
Water
pressure
(w/c=0.3)
Water
pressure
(w/c=0.5)
2
1
0
0.69
17.24
34.47
Curing pressure (MPa)
Figure 7.23: Tensile strength of Class H-II cement with different w/c ratios (age = 72 h)
7.4.3.2.2 Class H-P cement
Similar to Class H-I and H-II cements, specimens produced with Class H-P cement cured at
the various pressures were also found to have clean, flat fracture planes during both water
pressure tests and splitting tensile tests. Some representative pictures of fracture planes are
shown in Figure 7.24. The results suggest that depressurization at the rate of 0.345 MPa/min
probably did not cause damage to the specimens prepared with Class H-P cement.
188
Figure 7.24: Fracture planes of water pressure tests (left) and splitting tensile tests (right)
(Test H-P-3)
The system deformation behaviors during depressurization of tests conducted with Class HP cement (with a w/c ratio of 0.38) are presented in Figure 7.25. The measured system
deformation gradients of different tests converge reasonably well except for Test H-P-3, due to a
leakage problem that reduced the total amount of pressurizing water for this test. Figure 7.25 also
shows that the system bulk modulus (which takes into account the total system volume)
calculated from different tests exhibits much better agreements. The deformation behavior of
Class H-P cement was very similar to those of Class H-I and H-II cements presented previously.
For comparison, the result of Test H-I-4 (which happens to have nearly the same total system
volume as Test H-P-4 at the end of the curing period) is also included in the figure. Probably due
to the lower permeability of Class H-P cement, the flow rate (proportional to deformation
gradient) during depressurization seems to start at a lower value compared with Class H-I cement.
Correspondingly, the initial rapid increase in flow rate appears to last longer for Class H-P
cement, suggesting that a slightly higher threshold pressure difference between pore pressure and
surrounding water pressure is needed to achieve a sufficient flow rate to enter the steady state
(See section 7.4.2).
189
4.5
-0.45
System bulk modulus (GPa)
System deformation gradient (mL/MPa)
-0.4
-0.5
-0.55
-0.6
H-P-2
H-P-3
H-P-4
H-I-4
-0.65
-0.7
0
10
20
30
40
Pressure (MPa)
50
60
4
3.5
3
H-P-2
H-P-3
H-P-4
2.5
2
0
10
20
30
40
Pressure (MPa)
50
60
Figure 7.25: System deformation gradient and bulk modulus variations with pressure
(Class H-P cement, w/c = 0.38)
Figure 7.26 shows the tensile strength test results of Class H-P cement pastes with a w/c
ratio of 0.38 for different curing conditions. The coefficient of variation ranges between 0.06 and
0.22 for splitting tensile tests and between 0.05 and 0.19 for water pressure tests. Similar to the
test results of Class H-I and H-II cements, splitting tensile strength increases very little with with
increasing curing pressure with some random variations, while water-pressure tensile strength
increases relatively more significantly. The average splitting tensile strength of Class H-P
cement cured at ambient temperatures (which ranged between 2.75 and 3.10 MPa for different
curing pressures) were nearly identical to those of Class H-II cement with the same w/c ratio
under similar curing conditions (which ranged between 2.80 and 3.17 MPa, see Figure 7.22).
However, the average water-pressure tensile strength of Class H-P cement cured at ambient
temperatures was about 25% higher than that of Class H-II cement. At ambient temperatures, the
correlation factor between water pressure tensile strength and splitting tensile strength is
2.07±0.19 for Class H-P cement, which is much higher than that for Class H-I and H-II cements.
190
The results suggest that Class H-P cement has lower permeability than Class H-I and H-II
cements, which is consistent with the previous system deformation analysis.
Splitting
Water pressure
Tensile strength (MPa)
8
7
6
5
4
3
2
1
0
25.6°C
0.69MPa
22.2°C
17.2MPa
23.9°C
34.5MPa
26.1°C
51.7MPa
Curing condition
Figure 7.26: Effect of curing condition on the tensile strength of Class H-P cement
(age = 72 h, w/c = 0.38)
7.4.3.2.3 Class C cement
Similar to all Class H cements, specimens produced with Class C cement cured at various
pressures were also generally found to have clean, flat fracture planes during both water pressure
tests and splitting tensile tests. The results suggest that depressurization at the rate of 0.345
MPa/min is not likely to cause serious damage to the specimens prepared with Class C cement.
However, probably due to its relatively weak strength, the specimen produced with Class C
cement appeared to be more prone to fracture along the active seal, which sometimes led to
invalid water pressure test results.
191
The system deformation behaviors during depressurization of tests conducted with Class C
cement (with a w/c ratio of 0.56) are presented in Figure 7.27. It is difficult to directly compare
the test results with Class H cements due to the significantly different w/c ratio and degree of
hydration. Nevertheless, since the deformation behaviors of all Class H cements are similar, the
result of Test H-II-5-3 (H-II cement, w/c = 0.5) is plotted in the same figure to be used as a
reference. The deformation behavior of Class C cement was also found to be generally similar to
that of the Class H cements. However, the initial period of rapid increase in flow rate appears to
last longer than those of all Class H cements, suggesting that Class C cement has lower
permeability than all Class H cements and that a higher threshold pressure difference between
pore pressure and surrounding water pressure is needed to reach the steady state.
System deformation gradient (mL/MPa)
-0.35
-0.4
-0.45
-0.5
-0.55
C-2
C-3
C-4
H-II-5-3
-0.6
-0.65
-0.7
0
10
20
30
40
Pressure (MPa)
50
60
Figure 7.27: Variation of system deformation gradient with pressure (Class C cement, w/c = 0.56)
Figure 7.28 shows the tensile strength test results of Class C cement pastes with a w/c ratio
of 0.56 for different curing conditions. The coefficient of variation ranges between 0.15 and 0.31
192
for splitting tensile tests and between 0.04 and 0.15 for water pressure tests. Splitting tensile
strength is found to be influenced more significantly by lab temperature fluctuations than by
curing pressure and generally decreases with decreasing lab temperature. On the other hand,
water pressure tensile strength seems to increase slightly with increasing curing pressure despite
temperature fluctuations, consistent with test results of Class H cements. It is not clear what
caused the slight drop in water pressure tensile strength at the curing pressure of 51.7 MPa. At
ambient temperatures, the correlation factor between water pressure tensile strength and splitting
tensile strength is 2.60±0.42 for Class C cement, which is much higher than those for all class H
cements. The results suggest that Class C cement has lower permeability than all Class H
cements, which is consistent with the previous system deformation analysis.
Splitting
Water pressure
Tensile strength (MPa)
7
6
5
4
3
2
1
0
26.9°C
0.69MPa
27.5°C
17.2MPa
25°C
34.5MPa
25.6°C
51.7MPa
Curing condition
Figure 7.28: Effect of curing condition on the tensile strength of Class C cement
(age = 72 h, w/c = 0.56)
193
7.4.3.2.4 Class A cement
Unlike Class C and all Class H cements, specimens produced with Class A cement cured at
high pressures (≥ 34.5 MPa) had rough, irregular fracture planes from both water pressure tests
and splitting tensile tests, indicating pre-existing defects caused by depressurization. Figure 7.29
and Figure 7.30 show details of the fracture planes after water pressure tests and splitting tensile
tests, respectively. The results suggest that depressurization at the rate of 0.345 MPa/min can
cause serious damage to the specimens prepared with Class A cement when curing pressure is
34.5 MPa or higher.
Figure 7.29: Fracture planes of water pressure tests (Class A cement)
194
Figure 7.30: Fracture planes of splitting tensile tests (Class A cement)
The system deformation behaviors during depressurization of tests conducted with Class A
cement (with a w/c ratio of 0.46) are presented in Figure 7.31. The result of Test H-II-4 (Class
H-II cement, w/c = 0.38) is also included in the figure for comparison. The results of Tests A-3
and A-4 are found to be distinctively different from those of the other tests discussed previously
in that the plot of deformation gradient vs. pressure appears to be separated into two steady state
periods, which are characterized by two distinctively different slopes. The slope of the first
period is much higher than that of the second period and the latter is very similar to that of the
steady state period of Class H-II cement. The lower flow rate (proportional to deformation
gradient) and faster increase in flow rate during the first steady state period is probably
associated with a gradual increase in the pressure difference between the pore pressure and the
195
surrounding water pressure. These results suggest that Class A cement has lower permeability
than Class C and all Class H cements, whose deformation behaviors have been found to be
similar. The duration of the first steady state period appears to be independent of curing pressure
and corresponds to a measured pressure (i.e. surround water pressure) change of approximately
22 MPa. The sharp transition from one to another steady state is probably associated with a
critical pressure difference between the pore pressure and the surrounding water pressure that
results in the rupture of the pore structure. The specimens of Test A-2 (cured at 17.2 MPa) were
found to be not damaged probably because this critical pressure difference was not reached
during depressurization.
System deformation gradient (mL/MPa)
-0.35
-0.4
-0.45
-0.5
-0.55
-0.6
A-2
A-3
A-4
H-II-4
Potential points of damage
-0.65
-0.7
0
10
20
30
40
Pressure (MPa)
50
60
Figure 7.31: Variation of system deformation gradient with pressure (Class A cement, w/c = 0.46)
Figure 7.32 shows the tensile strength test results of Class A cement pastes with a w/c ratio
of 0.46 for different curing conditions. The coefficient of variation ranges between 0.08 and 0.16
for splitting tensile tests and between 0.06 and 0.45 for water pressure tests. According to the
previous analyses, there is no doubt that the specimens of Tests A-3 and A-4 were damaged
196
during the depressurization process. This damage caused a significant reduction in water
pressure tensile strength and a significant increase in the corresponding coefficient of variation
of the test results. However, splitting tensile strength, which was only found to generally
decrease with decreasing lab temperature, appeared to be unaffected by this damage. This may
be explained by the fact that water flows primarily in the radial direction during the
depressurization process, particularly in the middle section of the specimen. The radial flow is
more likely to result in cracks that are perpendicular to the loading direction of the water
pressure tests (i.e. axial direction) and parallel to the loading direction of the splitting tensile tests.
The results of Tests A-1 and A-2, whose specimens were not damaged during depressurization,
can be used to calculate the correlation factor between water pressure tensile strength and
splitting tensile strength. The average correlation factor was found to be 3.42 for Class A cement
at ambient temperatures, much higher than those for Class C and all Class H cements. The results
suggest that Class A cement has lower permeability than Class C and all Class H cements, which
is consistent with the previous system deformation analysis.
Tensile strength (MPa)
Splitting
Water pressure
8
7
6
5
4
3
2
1
0
24.4°C
0.69MPa
22.8°C
25°C
17.2MPa 34.5MPa
Curing condition
24.4°C
51.7MPa
Figure 7.32: Effect of curing condition on the tensile strength of Class A cement
(age = 72 h, w/c = 0.46)
197
7.4.3.2.5 Class G cement
Similar to Class A cement, specimens produced with Class G cement cured at high
pressures (≥ 34.5 MPa) also had rough, irregular fracture planes from both water pressure tests
and splitting tensile tests, indicating pre-existing defects caused by depressurization. Figure 7.33
and Figure 7.34 show pictures of fractured specimens after water pressure tests and splitting
tensile tests, respectively. The results suggest that depressurization at the rate of 0.345 MPa/min
can cause serious damage to specimens prepared with Class G cement when curing pressure is
34.5 MPa and higher.
Figure 7.33: Representative fractured specimens after water pressure tests (Class G cement)
198
Figure 7.34: Fractured specimens after splitting tensile tests (Class G cement)
The system deformation behaviors during depressurization of tests conducted with Class G
cement (with a w/c ratio of 0.44) are presented in Figure 7.35. The result of Test A-4 (Class A
cement, w/c = 0.46) is also included in the figure for comparison. The results of Tests G-3 and
G-4 are found to be somewhat similar to those of Test A-3 and A-4, except that the transitions
between the two steady state periods are characterized by a sharp increase followed by a sharp
decrease in flow rate (proportional to deformation gradient). These results suggest that Class G
cement also has lower permeability than Class C and all Class H cements. It is difficult to
determine the exact point of damage since transition from the first period to the second period
appears to occur over a period of time. The duration of the first steady state period (assuming it
199
ends at the peak) also appears to be independent of curing pressure and corresponds to a
measured pressure change of approximately 25 MPa, similar to that of Class A cement.
System deformation gradient (mL/MPa)
-0.35
-0.4
-0.45
-0.5
-0.55
-0.6
G-2
G-3
G-4
A-4
-0.65
Potential points of damage
-0.7
0
10
20
30
40
Pressure (MPa)
50
60
Figure 7.35: Variation of system deformation gradient with pressure (Class G cement, w/c = 0.44)
Test results of both Class A and Class G cement indicate that specimen damage during
depressurization is mainly induced by the low permeability of the cement paste, which does not
permit a sufficient flow rate for pore water to readily expand and flow out. As shown in Eq. (7.5),
this “sufficient flow rate” is approximately proportional to the pressure gradient. Therefore,
lowering the pressure gradient will help to lower the threshold flow rate and could potentially
avoid damaging the specimens during depressurization.
Figure 7.36 shows the tensile strength test results of Class G cement pastes with a w/c ratio
of 0.44 for different curing conditions. The coefficient of variation ranges between 0.10 and 0.34
for splitting tensile tests and between 0.07 and 0.45 for water pressure tests. Similar to the results
of Class A cement, specimen damage caused by depressurization was again found to only have a
200
significant effect on water pressure tensile strength. Splitting tensile strength again generally
decreased with decreasing lab temperature and was largely unaffected by such damage. The
results of Tests G-1 and G-2, whose specimens were not damaged during depressurization, can
be used to calculate the correlation factor between water pressure tensile strength and splitting
tensile strength. The average value was found to be 3.01 for Class G cement at ambient
temperatures, which is higher than those for Class C and all Class H cements. The results suggest
that Class G cement has lower permeability than Class C and all Class H cements, which is
consistent with the previous system deformation analysis.
Tensile strength (MPa)
Splitting
Water pressure
9
8
7
6
5
4
3
2
1
0
25°C
24.7°C
23.1°C
25°C
0.69MPa 17.2 Mpa 34.5MPa 51.7MPa
Curing condition
Figure 7.36: Effect of curing condition on the tensile strength of Class G cement
(age = 72 h, w/c = 0.44)
7.5 Summary
This chapter mainly investigates the correlations between the tensile strengths of cement
pastes measured by an innovative water pressure test method and by the traditional splitting
tensile test method. Specimens produced with different types of cement were cured under
201
different temperatures and pressures and tested at the age of 48 or 72 hours. Water pressure tests
were performed at both in-situ conditions and non in-situ conditions (depressurized) while
splitting tests were all performed after depressurization. The following conclusions can be drawn
from the test results:
1. For a cement paste with serious bleeding before setting, its final w/c ratio after hardening
increases slightly with increasing curing temperature due to faster setting, but is largely
independent of curing pressure. Correspondingly, the density of a hardened cement paste
decreases slightly with increasing curing temperature but varies very little with curing
pressure. These variations become smaller when bleeding is minimized.
2. The density and splitting tensile strength of a vertically cast cylindrical cement paste
specimen increases from top to bottom along the axial direction due to settling. The strength
of the middle section is most representative of the average strength of the specimen.
3. The splitting tensile strength of a 51mm-diameter cylindrical cement paste specimen
increases significantly with increasing bearing strip width adopted in the test. Using a relative
bearing strip with (b/D ratio) of approximately 0.08 appears to give a tensile strength closest
to that obtained from direct tension tests performed on briquette samples.
4. Splitting tensile strength of a cement paste generally increases with increasing curing
temperature, especially at early ages due to significantly increased total degree of hydration.
It increases very little with increasing curing pressure primarily because the increase in total
degree of hydration is relatively small.
5. The tensile strength of a cylindrical cement paste specimen cured at high temperatures and
pressures can be tested in-situ by the water pressure test method. The statistical scatter of test
results is similar to that of the traditional test methods, such as splitting tensile tests.
202
However, at a loading rate of 0.69 MPa/min is used, water pressure tensile strength of a
cement paste decreases with decreasing viscosity of water and increasing permeability of the
specimen. It should not be directly compared at different temperatures due to variations of
the viscosity of water with temperature. For specimens not damaged during depressurization,
the non in-situ water pressure tensile strength (i.e. tested with the same axial pressure)
increases slightly with increasing curing pressure, probably due to the combined effect of
increasing degree of hydration and decreasing specimen permeability. The in-situ water
pressure tensile strength (i.e. tested with different axial pressure) increases relatively more
significantly with increasing curing pressure primarily because the axial pressure of the
specimen increases with increasing curing pressure during the test.
6. Depressurization at a certain rate (0.345 MPa/min is used in this study) of cement pastes
cured at high pressures can cause serious damage depending on the permeability of the
specimen and the duration of the depressurization process. Decreasing specimen permeability
and increasing duration of the depressurization process both increase the specimen’s
susceptibility to damage.
7. Whether depressurized cement paste specimens originally cured at high pressure have preexisting defects can be determined by examining the fracture planes of the specimens after
mechanical tests. Specimens without such defects usually have clean, flat fracture planes
while those with defects tyically have rough, irregular fracture planes.
8. Whether cement paste specimens cured at high pressure are damaged during depressurization
can also be determined by examing the variation of the system deformation gradient with
pressure. A sharp slope change is typically observed in the plot at the potential damage point.
203
CHAPTER 8 : CONCLUSIONS AND FUTURE WORK
This dissertation describes a series of experimental and theoretical investigations related to
the fundamental hydration kinetics and mechanisms of Portland cement as well as the effects of
curing temperature and pressure on its various properties. This chapter summarizes the main
findings of this study by separating them into the following five sections: (1) a new chemical
shrinkage test method developed to evaluate cement hydration kinetics; (2) a new hydration
kinetics model proposed to explain the effect of curing temperature and pressure on cement
hydration; (3) a new explanation of the fundamental hydration mechanisms of Portland cement
and the corresponding model; (4) an innovative tensile strength test method that allows in-situ
determination of the tensile strength of cement; (5) an innovative method of evaluating the
damage mechanisms during depressurization of cement paste specimens cured under high
pressures.
8.1 New Chemical Shrinkage Test for Evaluating Cement Hydration Kinetics
•
When a curing pressure of 0.69 MPa or higher is applied to a cement paste during hydration,
total chemical shrinkage measured for a period of 72 hours is largely independent of
specimen thickness.
•
The chemical shrinkage test is an important alternative to the isothermal calorimetry test to
evaluate cement hydration kinetics. It has the advantage of allowing the effect of hydrostatic
curing pressure on hydration to be easily measured.
204
•
The normalized total chemical shrinkage of cement during hydration is approximately equal
to its degree of hydration. The normalization factor, CS0 (total chemical shrinkage at
complete hydration), maybe estimated using the following equation:
CS 0 = wn0 (vw − vn )
(8.1)
where wn0 is the total non-evaporable water at complete hydration, which can be estimated
using an empirical equation; vw and vn are the specific volumes of capillary water and nonevaporable water, respectively, which vary with curing condition and are estimated to be
0.988 and 0.752 cm3/g, respectively, at ambient condition.
•
The normalized total chemical shrinkage of cement during hydration correlates strongly with
the normalized cumulative heat evolution, especially for cements with low C3A content. The
normalization factor of the latter, H0 (total heat evolution at complete hydration), is believed
to be independent of curing temperature, while the normalization factor of the former, CS0, is
found to decrease approximately linearly with increasing curing temperature. The linear
reduction rate is found to be 0.63%, 0.66%, 0.59%, 0.75%, and 0.64% per °C for Class A, C,
G, H-P, and H-I cement, respectively. The ratio of H0 to CS0 ranges from 7500 to 8000 J/mL,
depending on the type of cement.
•
The hydration rates measured by chemical shrinkage and by heat evolution differ quite
significantly from each other during very early stages (before the acceleration period)
probably due to the imbalance between dissolution and precipitation rates during this
particular period.
•
Cement hydration kinetics measured by chemical shrinkage is less accurate than that
measured by heat evolution mainly because estimation of the normalization factor CS0
205
(which depends on curing condition) involves a lot more approximations than that of the
normalization factor H0 (which is invariant with curing condition).
8.2 Effect of Curing Temperature and Pressure on Cement Hydration
•
Due to the particular mechanism of cement hydration, the rate of hydration is strongly
dependent on the total amount of hydration products generated on the surface of cement
particles as well as in the inter-particle spaces. Therefore, the rate of hydration should be
expressed as a function of the degree of hydration to allow better understanding of the effects
of curing temperature and pressure on hydration kinetics. The increase in hydration rate due
to a curing condition change from a reference condition (Tr, Pr) to an arbitrary condition (T,
P) is represented by a more or less constant scale factor of C, which is especially accurate for
cements with simpler compositions and for relatively small curing condition changes.
•
The change in cement hydration rate (i.e. the scale factor C) as a result of the curing
condition change is related to the activation energy (Ea) and the activation volume (∆V‡) of
the cement by the following equation:
 E  1 1  ∆V ‡  Pr P  
C (T , P ) = exp  a  −  +
 −  
 R T T
R
 T T 

  r
(8.2)
where R is the gas constant. Since cement is a composite material that consists of many
different phases, which hydrate at different rates and have different sensitivities to curing
temperature and pressure, the scale factor (as well as the apparent activation energy and the
apparent activation volume of the cement) varies with the progress of hydration. The
variations become smaller for cements with simpler compositions and for relatively small
curing condition changes.
206
•
The hydration kinetics of cement may be represented by three different types of curves:
degree of hydration vs. time, rate of hydration vs. time and rate of hydration vs. degree of
hydration. For each of these three types of curves, test results obtained at different curing
conditions remain by and large invariant if properly transformed with a set of scaling factors.
Therefore, the experimental hydration kinetics curve at one curing condition can be used to
predict that of another curing condition using the scale factor C (Table 4.3). The accuracy of
prediction is better for cements with simpler compositions and for relatively small curing
condition changes. If the properties (activation energy and activation volume) of the cement
are known, then C can be simply calculated from Eq. (8.2). Otherwise, C may be determined
by trial and error such that test results from two different curing conditions have the best
agreement. The obtained value of C can then be used to estimate the activation energy and
the activation volume of the cement.
•
There are several other methods that can be used to estimate the scale factors. Probably the
most straightforward way is using the peak hydration rates at different curing conditions as
they are usually associated with the same degree of hydration. However, this method may
result in significant errors when the hydration peaks of C3S and C3A superimpose on each
other, which typically occurs at high curing temperatures.
•
The scale factor due to a pressure increase from 0.69 MPa to 51 MPa ranges from 1.56 to
1.82 for different cements based on chemical shrinkage test data, suggesting that the
hydration rate is increased by 56% to 82%. The scale factor due to a temperature increase
from 25 °C to 40 °C ranges from 2.15 to 2.3 for different cements based on heat evolution
data, suggesting that the hydration rate is increased by 115% to 130%. Therefore, cement
hydration is much more sensitive to temperature changes than pressure changes.
207
8.3 A New Explanation of Cement Hydration Mechanisms
Due to uncertainties about the detailed mechanisms of cement hydration, few existing
models can accurately reproduce the entire hydration kinetics curves (both the integral curve and
the derivative curve) despite decades of investigations. A three-parameter model proposed in this
study is found to provide exceptional fits to experimental data of both C3S hydration in dilute
suspensions and cement paste hydration for a curing period up to 72 hours. It is very difficult to
directly observe or measure cement hydration mechanisms. Our understanding of the
mechanisms is mainly achieved through fitting models that are developed based on a set of
assumptions with experimental data. Unfortunately, a good fit with experimental data does not
necessarily guarantee that the assumptions made in the model are correct. Therefore, the
conclusions drawn in this section need to be further verified when new experimental techniques
become available.
•
C3S hydration in dilute suspensions follows a similar mechanism as cement paste hydration,
both of which can be accurately modeled by a combined cement hydration model that
connects a nucleation and growth controlled mechanism with a diffusion controlled
mechanism.
•
During the nucleation and growth stage of hydration, nucleation of hydration products
primarily occurs before the acceleration stage of hydration and may be approximated as a site
saturation condition (i.e. all nuclei are instantly formed at the start of the reaction). Nuclei are
mainly formed on the surface of cement particles and grow at two constant, but different
rates: one parallel to the particle surface, the other one perpendicular to the particle surface.
•
Both the parallel and the perpendicular growth rates of the nuclei vary significantly with
curing condition. Their dependencies on curing temperature and pressure are associated with
208
the activation energy and the activation volume of the cement, respectively, and can be
modeled by chemical kinetics theory.
•
Hydration of each cement particle becomes diffusion controlled as soon as its surface is
completely covered by hydration products. Such transition occurs earlier for smaller particles
than larger ones. Therefore, for a cement paste sample with multiple particle sizes, the
transition of rate controlling mechanisms occurs through a period of time, rather than at a
fixed time. The transition period roughly corresponds to the deceleration stage of hydration.
•
Only inner hydration products (i.e. those formed in the space between the anhydrous cement
particle and the hydration products formed during the nucleation and growth stage) act as the
diffusion barrier during the diffusion controlled stage of hydration.
•
The dependence of the diffusion constant on curing temperature is similar to those of nuclei
growth rates while its dependence on curing pressure appears to be too small to be modeled
with the available test data.
•
For cement paste hydration, inter-particle interactions are minimal during the nucleation and
growth stage of hydration, but become significant during the diffusion controlled stage of
hydration. Nuclei growth rates are largely independent of w/c ratio while the apparent
diffusion constant increases with increasing w/c ratio due to increasing inter-particle
distances that reduce interactions.
8.4 Water Pressure Tensile Test of Oilwell Cement
•
The tensile strength of a cylindrical cement paste specimen cured at high temperatures and
pressures can be tested in-situ by either increasing the annulus pressure or decreasing the end
pressure of the specimen. The difference between the annulus pressure and the end pressure
209
of the specimen at the time of fracture is defined as the in-situ water pressure tensile strength.
The statistical scatter of water pressure test results is similar to that of the traditional test
methods, such as splitting tensile tests.
•
During a typical in-situ water pressure test, the specimen is under a triaxial state of loading.
The in-situ water pressure tensile strength of the specimen increases with increasing curing
pressure partially because the axial pressure of the specimen increases with increasing curing
pressure during the test.
•
While the particular mechanism of fluid pressure tests is still uncertain, it appears that the
correlation factor (α = ffpt/fst) between the fluid pressure tensile strength (ffpt) and the splitting
tensile strength (fst) of a cement paste decreases with decreasing loading rate, decreasing fluid
viscosity, as well as increasing permeability of the specimen. However, the ratio is always
larger or equal to 1 (approximately), suggesting that further changes of the three influencing
parameters will not affect test results when a threshold value is reached.
•
A loading rate of 0.69 MPa/min is found to be far above the threshold value for all the
cements used in this study. At ambient temperatures, the average correlation factor (α)
between the fluid pressure tensile strength and the splitting tensile strength is determined to
be 1.56, 2.07, 2.61, 3.01, and 3.42, for Class H-II, H-P, C, G, and A cement, respectively,
suggesting that Class A cement has the lowest permeability.
•
Since both the viscosity of water and the permeability of cement paste vary with curing
temperature, the water pressure tensile strength of cement pastes cannot be directly compared
at different curing temperatures.
•
For specimens not damaged during depressurization, the non in-situ water pressure tensile
strength (i.e. tested with the same axial pressure) increases slightly with increasing curing
210
pressure probably due to the combined effect of increasing degree of hydration and
decreasing specimen permeability.
8.5 Damage Mechanism of Cement Paste Specimen during Depressurization
•
Depressurizing cement pastes cured at high pressures at a certain rate can cause serious
damage to the specimens depending on the permeability of the specimen and the duration of
the depressurization process. Decreasing specimen permeability and increasing duration of
the depressurization process both increase the specimen’s susceptibility to damage.
•
Analyzing the deformation behavior of a system (deformation gradient vs. pressure) that
consists of cement paste specimens and pressurizing water can serve as an important method
to evaluate the damage mechanism of the specimens during depressurization.
•
A hardened cement paste with relatively high permeability typically has a similar
deformation behavior as a cement slurry during depressurization. However, a small period of
relatively rapid increase in the system deformation gradient is usually observed for hardened
cement paste. This period is believed to be associated with a pressure difference build up
process during which the pore pressure decreases slower than the surrounding water pressure.
The increased pressure difference would enable a steady outflow of pore water to allow
release of inside pressure. However, the pressure build up period may last for a very long
time due to low permeability of the specimen and the high pressure difference may
eventually damage the specimen. The occurrence of such damage is usually characterized by
a sharp slope change in the plot of deformation gradient vs. pressure.
211
8.6 Recommendations for Future Research
•
As mentioned repeatedly in this dissertation, an important drawback of the newly designed
apparatus is the lack of a precise temperature control scheme. Therefore, test results
presented in this study are not obtained at truly isothermal conditions and the recorded
temperatures of the tests are not very accurate. Since cement hydration is very sensitive to
temperature changes, a better temperature control scheme needs to be developed.
•
The hydration models developed in this study are only applied to experimental data at early
ages (typically ≤ 72 h), their applicabilites to experimental data at later ages and potential
modifications of the models should be further investigated.
•
Fitted results of the three-parameter model developed in this study suggest that the number of
nuclei per unit surface area varies with cement particle size. Further studies are needed to
establish a more accurate relationship.
•
It is shown in Chapter 6 that it is theoretically possible to continuously measure the
individual hydration kinetics of different phases for cement that consists of only two phases.
This should be further explored experimentally as it may improve our understanding of the
interactions between different phases.
•
It is shown in Chapter 7 that theoretically the effect of curing temperature and pressure on the
time evolution of many physical and mechanical properties of cement can be modeled in the
same way as hydration kinetics using the one-parameter model proposed in this study.
Further experimental investigation of this concept is recommended.
•
It is stated in Chapter 7 that lowering the pressure gradient during depressurization could
potentially reduce the risk of damaging the specimens. Further experimental proof of this
hypotheis is recommended.
212
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